tag:blogger.com,1999:blog-281729052024-03-17T22:59:48.333-04:00(x, why?)A weblog of math-related cartoons and images, math cartoons, school life humor and general math discussion. Or whatever else comes to mind. And that's when it gets weird. Geek-based humor will occur.(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.comBlogger3030125tag:blogger.com,1999:blog-28172905.post-11026981907441830162024-03-14T12:47:00.001-04:002024-03-14T12:47:34.921-04:00Pi Day 2024<font size="2">(Click on the comic if you can't see the full image.)</font><br />
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<iii src="http://mrburkemath.net/xwhy/images/1776 .png" title="The co-sin as well!"></iii>
</aa>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9zczSUnmZswN4GqzdWg-pYNvk4Quo9rYjdqQ1ItR0rnl7RONjPwv_8hQ1plCBXwda5bpcXa9zKuuDjTlqmvSw2MWQ5YeyraBA4CgkIMjiTHNPPJEyMMWmKVytF3Rxj1ZlEBCXoZ_nGworN1GHCIMNENdE5sGGdOBfpKF3HZZ_4SQspt-jXGb_6A/s1600/piday2024-sin-pi-div-4.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="550" data-original-width="600" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9zczSUnmZswN4GqzdWg-pYNvk4Quo9rYjdqQ1ItR0rnl7RONjPwv_8hQ1plCBXwda5bpcXa9zKuuDjTlqmvSw2MWQ5YeyraBA4CgkIMjiTHNPPJEyMMWmKVytF3Rxj1ZlEBCXoZ_nGworN1GHCIMNENdE5sGGdOBfpKF3HZZ_4SQspt-jXGb_6A/s1600/piday2024-sin-pi-div-4.png"/></a></div>
<font color="FF8844">(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).</font>
</center>
<font size=4>
<P>The co-sin as well!</P>
<P>I know I usually save these guys for Talk Like A Priate Day, and use Sherlock Pi on Pi Day (3/14) but I decided to shake things up a little.</P>
<P>I've been getting a lot of suggestions for using the Sin of Pie is 0, which is amusing, but it's not my joke, so unless I can work it into my characters, I can't use it. (I could, but I choose not to.)</P>
<P>On the other hand, if you quarter pi, then the sin is radical 2 over 2. And so is the cos. </P>
<P>And your tan will be the one.</P>
</font>
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
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<br /><br /><i>Come back often for more funny math and geeky comics.</i>
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<br />(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-2564445786745067442024-03-12T10:43:00.003-04:002024-03-12T10:43:43.418-04:00January 2024 Algebra 2 Regents, Part III<font size=4 face="Times New Roman">
<BR><P>This exam was adminstered in January 2024. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>Algebra 2 January 2024</h3></u></em>
<P>Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>33. A researcher wants to determine if nut allergies and milk allergies are related to each other. The
researcher surveyed 1500 people and asked them if they are allergic to nuts or milk. The survey
results are summarized in the table below.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3D89QVqFPOeQmfdQTXUoCPRoQIB0bqkNUQe4Ewf5COE0XSel6PDcF9PFLEyIZflNm56blbuwHE-sKTq9WfAlF9G4L4QfyO4lWLn8FEMZ8T7AnJVaPGpZDmI2rcdGZSLdp4Vz2NjyW6PRWFBGYqiBfwUJ_EyUNRsZe1B10GbkBzxgVOc4xch8Oaw/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="259" data-original-width="451" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3D89QVqFPOeQmfdQTXUoCPRoQIB0bqkNUQe4Ewf5COE0XSel6PDcF9PFLEyIZflNm56blbuwHE-sKTq9WfAlF9G4L4QfyO4lWLn8FEMZ8T7AnJVaPGpZDmI2rcdGZSLdp4Vz2NjyW6PRWFBGYqiBfwUJ_EyUNRsZe1B10GbkBzxgVOc4xch8Oaw/s1600/temp00.png"/></a></div>
<BR>Determine the probability that a randomly selected survey respondent is allergic to milk
<BR>Determine the probability that a randomly selected survey respondent is allergic to milk, given
that the person is allergic to nuts.
<BR>Based on the survey data, determine whether nut allergies and milk allergies are independent
events. Justify your answer.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Add up the rows and columns. You will see that that it adds to 1500, as stated in the problem.
<P>There are 45 respondents out of 1500 who are allergic to milk, so the probability is 45/1500. (You don't need to simplify the fraction.)
<P>There are 15 people who are allergic to nuts, and of those, 3 are also allergic to milk, so the probability is 3/15.</P>
<P>If the nut allergies and milk allergies are independent, then the previous two answers would be the same because P(A) would have to be equal to P(A|B). However, 45/1500 = 0.03 and 3/15 = 0.2. So the events are not independent. </P>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLsAT4GW4OGflpa8IZZuK75LKdkNPPZwDtcjVKHbQ9kd7ZFXWY2nZfipX1ZrWW0zIFiIw09Na7HWRVMiEuBlSMzM26dlCQi44na3o-MGJjLA2n2gUjxp2dnbwhgtVMO1J7B5ZVmTad6XRSzoKGs7glp63F2dSrIFeayoQCsuCqr_NlKMxOEVtK7w/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="452" data-original-width="497" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLsAT4GW4OGflpa8IZZuK75LKdkNPPZwDtcjVKHbQ9kd7ZFXWY2nZfipX1ZrWW0zIFiIw09Na7HWRVMiEuBlSMzM26dlCQi44na3o-MGJjLA2n2gUjxp2dnbwhgtVMO1J7B5ZVmTad6XRSzoKGs7glp63F2dSrIFeayoQCsuCqr_NlKMxOEVtK7w/s1600/temp00.png"/></a></div>
<P>
<BR><BR><BR>
<BR><em>34. Algebraically solve for x: 2x = 6 + 2√(x - 1)
</em>
<P>
<P><B>Answer: </B> </P>
<br>Isolate the radical. Then square both sides. Finally, solve the quadratic equation.
<CENTER><P>2x = 6 + 2√(x - 1)<BR>
2x - 6 = 2√(x - 1)
<BR> x - 3 = √(x - 1)
<BR>(x - 3)<sup>2</sup> = (√(x - 1))<sup>2</sup>
<BR>x<sup>2</sup> - 6x + 9 = x - 1
<BR>x<sup>2</sup> - 7x + 10 = 0
<BR>(x - 5)(x - 2) = 0
<BR>x - 5 = 0 or x - 2 = 0
<BR>x = 5 or x = 2</P>
</CENTER>
<P>Throw out x = 2 as extraneous because 2(2) =/= 6 + 2√(2-1).</P>
<P>The only solution is x = 5.</P>
<BR><BR><BR>
<BR><em>35. During the summer, Adam saved $4000 and Betty saved $3500. Adam deposited his money in
Bank A at an annual rate of 2.4% compounded monthly. Betty deposited her money in Bank B at
an annual rate of 4% compounded quarterly. Write two functions that represent the value of each
account after t years if no other deposits or withdrawals are made, where Adam’s account value is
represented by A(t), and Betty’s by B(t).
<P>Using technology, determine, to the nearest tenth of a year, how long it will take for the two
accounts to have the same amount of money in them. Justify your answer.</P>
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Write the functions A(t) and B(t) using the given initial amounts and rates. Note that that A is compounded monthly, so 0.024 will be divided by 12 and the exponent will be multiplied by 12. Likewise, B is compounded quarterly, so 0.04 will be divided by 4 and the exponent will be multiplied by 4.
<CENTER><P>A(t) = 4000(1 + 0.024/12)<sup>12t</sup> <BR>or A(t) = 4000(1.002)<sup>12t</sup> </P>
<P>B(t) = 3500(1 + 0.04/4)<sup>12t</sup> <BR>or B(t) = 3500(1.01)<sup>4t</sup> </P>
</CENTER>
<P>"Using technology" means that you can graph the two functions to see when they intersect rather than writing an equation and solving. Put the two equations into your graphing calculator. Graph and trace the functions, or look at the table of values, setting the calculator to show every 0.1 value of x.</P>
<P>At t=8.4, A(8.4) = 4892.40 and B(8.4) = 4889.50, a difference of $3.10, which is the smallest difference to the nearest tenth of year.</P>
<BR><BR><BR>
<BR><em>36. <div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiWKuoIKjLQdBtlz77fTBD-fGCQfICF9zVO-xgIF2E9ZE5slFBUOLEtRLrNLmgaSs-USQz0EgLnzH8ckvSAtHZ6CvIL341fOPaF1wQ5YWnnYwNq0R4O8zH5Hrn4R6bNBBH5u0GFbARoL2b2D80lpbMj8zOyucHXWugjJOB_i9U02XC2Ns-s6ocbA/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="419" data-original-width="462" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiWKuoIKjLQdBtlz77fTBD-fGCQfICF9zVO-xgIF2E9ZE5slFBUOLEtRLrNLmgaSs-USQz0EgLnzH8ckvSAtHZ6CvIL341fOPaF1wQ5YWnnYwNq0R4O8zH5Hrn4R6bNBBH5u0GFbARoL2b2D80lpbMj8zOyucHXWugjJOB_i9U02XC2Ns-s6ocbA/s1600/temp00.png"/></a></div>On the graph below, draw at least one complete cycle of a sine graph passing through point (0,2)
that has an amplitude of 3, a period of p, and a midline at y = 2.
<BR>Based on your graph, state an interval in which the graph is increasing.
</em>
<P>
<P><B>Answer: </B> </P>
<BR>The period of π instead of 2π means that means we need sin(2x) instead of sin(x). The amplitude of 3 is a mulitplier in front of the function and the midline of 2 is an addition after the function.
<P>y = 3 sin(2x) + 2</P>
<P>The graph would look like the one below. The y-intercept is at (0,2). The maximum value is 5 and the minimum is -1.</P>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhWxCtDEGR4zdtnYh1H_HsFGYdbNHbuAob-ybwLUiubzOvVoyxcSwFaaC1wrj6wv5Ad9o-Hj3TuHgE_LPDGd-G6o2QpDmfpSJKjWTYwS7a756nSLxr_6PZPBdpHcXEBnX7Cu06sPr4pvE3GZs6U3JVWQhYqNpj90tb1b3dO8mPqUj4PT2joMFDgw/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="419" data-original-width="432" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhWxCtDEGR4zdtnYh1H_HsFGYdbNHbuAob-ybwLUiubzOvVoyxcSwFaaC1wrj6wv5Ad9o-Hj3TuHgE_LPDGd-G6o2QpDmfpSJKjWTYwS7a756nSLxr_6PZPBdpHcXEBnX7Cu06sPr4pvE3GZs6U3JVWQhYqNpj90tb1b3dO8mPqUj4PT2joMFDgw/s1600/temp00.png"/></a></div>
<P>
<P>One interval where it is increasing would be from 3π/4 to 5π/4.</P>
<P>
<BR><BR><BR>
End of Part III
<P>How did you do?</P>
<BR><BR><BR>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-68897309708440997972024-03-08T13:06:00.001-05:002024-03-08T13:06:07.181-05:00Arctan<font size="2">(Click on the comic if you can't see the full image.)</font><br />
<center>
<aa href="http://xwhy.comicgenesis.com/d/20240308.html">
<iii src="http://mrburkemath.net/xwhy/images/1776 .png" title="I know, the Ham joke was a little cheesy. "></iii>
</aa>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDi1CIKjDEXkubgTZUwFPWkHK7va0_mh95lYofZqci7K895BIfySyXEgkj6WVBjEzTenu-t249hZCT5uhZCp5P-cWmlUfLVEheM4qBSP1f66WYlWBva1BaLgriVpNzyxlwG9LdYrtNrY49zxLU9YKKq7wO5_F8VLjZhWRGdxYXzbn7PMnsCNMngQ/s1600/noah06.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="400" data-original-width="650" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDi1CIKjDEXkubgTZUwFPWkHK7va0_mh95lYofZqci7K895BIfySyXEgkj6WVBjEzTenu-t249hZCT5uhZCp5P-cWmlUfLVEheM4qBSP1f66WYlWBva1BaLgriVpNzyxlwG9LdYrtNrY49zxLU9YKKq7wO5_F8VLjZhWRGdxYXzbn7PMnsCNMngQ/s1600/noah06.png"/></a></div>
<font color="FF7744">(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).</font>
</center>
<font size=4>
<P>I know, the Ham joke was a little cheesy.</P>
<P>I haven't used Noah and his sons in quite a while. They were one of the early themes, and I thought I'd revisit them as we closed in on comic #2000.</P>
<P>Also, as anyone with fair skin can tell you, you can get burned on cloudy days. UV rays don't care about clouds. </P>
<P>Since I have such fair skin, I tend to stay covered up even in the summer, so I have more of an inverse tan.</P>
</font>
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
<br /><br /><i>Come back often for more funny math and geeky comics.</i>
<center>
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<br />(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-13577452174457129242024-02-28T11:20:00.000-05:002024-02-28T11:20:14.489-05:00School Life #40: Beagle<font size="2">(Click on the comic if you can't see the full image.)</font><br />
<center>
<aa href="http://xwhy.comicgenesis.com/d/20240228.html">
<iii src="http://mrburkemath.net/xwhy/images/1776 .png" title="When you give all you have to give, that's giving 100%. The math checks. "></iii>
</aa>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJGTMgN-Pmu3dVGJQUcf7C4NTPFwuMdCQeozTxr2i2DbOegpUZyODIcgV8L5ybMMaCbBmkK1hF6isPgxRu26yZwu2n30XW96A6MyxUt6cPg91dBfzq8iAtSDPlK73JFa2duaj-278VvAMkhTj_RDflO9YLX5vtAkRRMfOgWHx80M-37CCn_F-1sQ/s1600/schoollife42-beagle.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="350" data-original-width="1000" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJGTMgN-Pmu3dVGJQUcf7C4NTPFwuMdCQeozTxr2i2DbOegpUZyODIcgV8L5ybMMaCbBmkK1hF6isPgxRu26yZwu2n30XW96A6MyxUt6cPg91dBfzq8iAtSDPlK73JFa2duaj-278VvAMkhTj_RDflO9YLX5vtAkRRMfOgWHx80M-37CCn_F-1sQ/s1600/schoollife42-beagle.png"/></a></div>
<font color="FF6644">(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).</font>
</center>
<font size=4>
<P>Everyone know Charlie Brown and Snoopy went to the Moon, not to the Galápagos!</P>
<P>This one has been in the pipeline for a while, waiting to be done. Problem was I forget about it a couple of times, but the good ones always come back.</P>
<P>Sometimes the bad ones do, too, like a bad (knock, knock, knock) Penny.</P>
</font>
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
<br /><br /><i>Come back often for more funny math and geeky comics.</i>
<center>
<br /><br /><a href="http://xwhy.comicgenesis.com/"><img src="http://mrburkemath.net/images/banner.png" /></a></center>
<br />(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-51464426647867582362024-02-19T10:41:00.000-05:002024-02-19T10:41:38.503-05:00January 2024 Algebra 2, Part II<font size=4 face="Times New Roman">
<BR>
<P>This exam was adminstered in January 2024. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>Algebra 2 January 2024</h3></u></em>
<P>Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>25. Factor the expression x<sup>3</sup> + 4x<sup>2</sup> - 9x - 36 completely.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Factor by grouping, and then factor the quadratic you get after the first step.
<P>There are two ways to group, and either should work in any question of this kind.</P>
<CENTER>
<P>x<sup>3</sup> + 4x<sup>2</sup> - 9x - 36
<br>(x<sup>3</sup> + 4x<sup>2</sup>) - (9x + 36)
<br>x<sup>2</sup>(x + 4) - 9(x + 4)
<br>(x<sup>2</sup> - 9)(x + 4)
<br>(x + 3)(x - 3)(x + 4)
</P></CENTER>
<P>You can also switch the two middle terms around. This is just the way I learned it, so I usually do it, especially if it helps me avoid factoring out a minus sign.</P>
<CENTER>
<P>x<sup>3</sup> - 9x + 4x<sup>2</sup> - 36
<br>(x<sup>3</sup> - 9x) + (4x<sup>2</sup> - 36)
<br>x(x<sup>2</sup> - 9) + 4(x<sup>2</sup> - 9)
<br>(x + 4)(x<sup>2</sup> - 9)
<br>(x + 4)(x + 3)(x - 3)
</P></CENTER>
<P>Note: This was <I>Very Similar</I> to Question 25 on the Auguest 2023 Regents. Right down to the (x + 3)(x - 3).</P>
<BR><BR><BR>
<BR><em>26. Determine if x + 4 is a factor of 2x<sup>3</sup> + 10x<sup>2</sup> + 4x - 16. Explain your answer.
</em>
<P>
<P><B>Answer: </B> </P>
<br>If (x + 4) is a factor of the polynomial, then the value of the polynomial must be 0 when x = -4.
<P>2(-4)<sup>3</sup> + 10(-4)<sup>2</sup> + 4(-4) - 16 = 0</P>
<P>Since the expression is equal to zero when x = -4, then (x + 4) must be a factor.</P>
<P>You could also solve this using polynomial division.</P>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlCRAKPEpHyuKzv0wO7BOiuLWnfXluuTQJok2xuk9lCe1Zxaf7vrOrKHYAB5yCF83xZVWfcsgFKiZt4IDIPmG-UPdobiLaPSqsmlfP4urFcGkWLMatDkdSH7gG1s1e8cAcWgX-8SsVzuaQHyqyW8erG2e1l_1whubHWmHU7VZOSJoLSZRR1CSdnA/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="194" data-original-width="265" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlCRAKPEpHyuKzv0wO7BOiuLWnfXluuTQJok2xuk9lCe1Zxaf7vrOrKHYAB5yCF83xZVWfcsgFKiZt4IDIPmG-UPdobiLaPSqsmlfP4urFcGkWLMatDkdSH7gG1s1e8cAcWgX-8SsVzuaQHyqyW8erG2e1l_1whubHWmHU7VZOSJoLSZRR1CSdnA/s1600/temp00.png"/></a></div>
<P>(x + 4) divides evenly, with no remainder, so it is a factor.</P>
<BR><BR><BR>
<BR><em>27. An initial investment of $1000 reaches a value, V(t), according to the model V(t) = 1000(1.01)<sup>4t</sup>,
where t is the time in years.
<BR>
Determine the average rate of change, to the nearest dollar per year, of this investment from year 2 to year 7.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Calculate V(7) and V(2). Subtract them and divide by 7 - 2, which is 5. You are looking for the rate of change (or slope, if you prefer).
<P>V(7) = 1000(1.01)<sup>4(7)</sup> = 1321.29</P>
<P>V(2) = 1000(1.01)<sup>4(2)</sup> = 1082.86</P>
<P>Rate of change = (1321.29 - 1082.86) / 5 = 47.686, which is $48 to the nearest dollar.</P>
<BR><BR><BR>
<BR><em>28. When ( 1 / ∛(y<sup>2</sup>) ) y<sup>4</sup> is written in the form y<sup>n</sup>, what is the value of n? Justify your answer.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Use the laws of exponents to change the radical into a fraction. The combine the terms.
<CENTER><P>( 1 / ∛(y<sup>2</sup>) ) y<sup>4</sup> <BR>( 1 / (y<sup>2/3</sup>) y<sup>4</sup>
<BR>(y<sup>-2/3</sup>) y<sup>4</sup>
<BR>y<sup>10/3</sup>
</CENTER>
<P>n = 10/3.</P>
<BR><BR><BR>
<BR><em>29. The heights of the members of a ski club are normally distributed. The average height is 64.7
inches with a standard deviation of 4.3 inches. Determine the percentage of club members, to the
nearest percent, who are between 67 inches and 72 inches tall.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>They don't use the chart with the normal distribution and all the standard deviations marked off any more. They just assume that you have and will use a calculator for this.
<P>You need to use the normalcdf function.</P>
<P>Enter the command normalcdf(67,72,64.7,4.3) and you will get .2515... or 25%.</P>
<P>All of the numbers that go into the command are in the question. Lower bound, upper bound, median, standard deviation.</P>
<BR><BR><BR>
<BR><em>30. The explicit formula a<sub>n</sub> = 6 + 6n represents the number of seats in each row in a movie theater,
where n represents the row number. Rewrite this formula in recursive form.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>A recursive function needs an initial value (a<sub>1</sub>) and an equation for a<sub>n</sub> is terms of a<sub>n-1</sub>.
<P>The inition value a<sub>1</sub> = 12.</P><P>Then a<sub>n</sub> = a<sub>n-1</sub> + 6, because the common difference (rate of change) is 6.</P>
<BR><BR><BR>
<BR><em>31.Write (2xi<sup>3</sup> - 3y)<sup>2</sup>) in simplest form.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Square the binomial, substitute the powers of <i>i</i>, and Combine Like Terms.
<CENTER><P>(2xi<sup>3</sup> - 3y)<sup>2</sup>)</P>
<P>(2xi<sup>3</sup> - 3y)(2xi<sup>3</sup> - 3y)</P>
<P>4x<sup>2</sup>i<sup>6</sup> - 6xyi<sup>3</sup> - 6xyi<sup>3</sup> + 9y<sup>2</sup></P>
<P>-4x<sup>2</sup> - 12xyi<sup>3</sup> + 9y<sup>2</sup></P>
<P>-4x<sup>2</sup> + 12xyi + 9y<sup>2</sup></P>
</CENTER>
<BR><BR><BR>
<BR><em>32. A survey was given to 1250 randomly selected high school students at the end of their junior year.
The survey offered four post-graduation options: two-year college, four-year college, military, or
work. Of the 1250 responses, 475 chose a four-year college. State one possible conclusion that can
be made about the population of high school juniors, based on this survey
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>This seems almost too simple a problem. If you divide 475/1250, you get .38 or 38%.
<P>One conclusion you can draw is that the population of high school juniors that would chose a four-year college would probably be about 38% and 62% would choose a different option. </P>
<BR><BR><BR>
End of Part II
<P>How did you do?</P>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-84631368203577051242024-02-16T13:32:00.002-05:002024-02-16T13:32:57.436-05:00January 2024 Geometry Regents Part IV
<BR><P>This exam was adminstered in January 2024. </P>
<!-- <BR>The answers to Part I can be found <a href="https://mrburkemath.blogspot.com/2019/04/january-2019-common-core-geometry.html">here</a>
<BR>The answers to Parts III and IV can be found <a href="https://mrburkemath.blogspot.com/2019/03/january-2019-common-core-geometry.html">here</a> -->
<font size=4 face=helvetica>
<P><h3>January 2024 Geometry, Part IV</h3>
<P>A correct answer is worth 6 credits. Partial credit can be given for correct statements in the proof.
<BR><BR><BR><P><strong>35.</strong> <em>Quadrilateral MATH has vertices with coordinates M(-1,7), A(3,5), T(2,-7), and H(-6,-3).
<BR>Prove that quadrilateral MATH is a trapezoid.
<BR>[The use of the set of axes on the next page is optional.]
<P>State the coordinates of point Y such that point A is the midpoint of MY.
<P>Prove that quadrilateral MYTH is a rectangle. [The use of the set of axes below is optional.]
</em>
<P><b>Answer: </b> </P>
<BR>You don't neeed to use the grid but it may be helpful for visualizing. If you do use it, you still need to answer the questions fully and completely. You can't rely on whatever is in the grid to be sufficient.
<P>To show that MATH is a trapezoid, you have to that there is one pair of parallel sides. You can do this by find the slopes of the four sides. This is easy if you graph it because you can count boxes without worrying about subtracting signed numbers (if that's a problem for you). Only two of the sides will be the same and the other two will be different.
<P>Slope MA (7 - 5) / (-1 - 3) = 2/-4 = -1/2</P>
<P>Slope AT = (-7 - 5) / (2 - 3) = -12/-1 = 12</P>
<P>Slope TH = (-3 - -7) / (-6 - 2) = 4/-8 = -1/2</P>
<P>Slope HM = (7 - -3) / (-1 - -6) = 10/5 = 2</P>
<P>Since there is one pair of parallel sides, the quadrilateral is a trapezoid.</P>
<P>To find point Y, find the change in x-value and y-value fro M to A and add those numbers again to get Y.</P>
<P>M(-1,7) -> A(3,5) is a translation of +4,-2. Y(3+4,5-2) = Y(7,3)
<BR>This was worth a point even if you didn't show work.</P>
<P>To show that MYTH is a rectangle, you could show that the opposite sides are parallel (same slope) and that two consecutive sides are parallel (slopes are inverse reciprocals).</P>
<P>Slope MY (7 - 3) / (-1 - 7) = 4/-8 = -1/2</P>
<P>Slope YT = (-7 - 3) / (2 - 7) = -10/-5 = 2</P>
<P>You previously found:
<br>Slope TH = (-3 - -7) / (-6 - 2) = 4/-8 = -1/2
<br>Slope HM = (7 - -3) / (-1 - -6) = 10/5 = 2</P>
<P>Opposite sides are parallel, so it is a parallelogram.</P>
<P>(-1/2)(1) = -1. MY is perpendicular to YT, so angle Y is a right angle. Therefore, MYTH is a rectangle.</P>
<BR><BR><BR>
<P><B>End of Exam</B>
<P>How did you do?
<P>Questions, comments and corrections welcome.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com2tag:blogger.com,1999:blog-28172905.post-73721381455824120162024-02-16T13:17:00.000-05:002024-02-16T13:17:36.166-05:00January 2024 Algebra 1 Regents Part IV<font size=4 face="Times New Roman">
<BR><BR><P>This exam was adminstered in January 2024. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>January 2024</h3></u></em>
<P>Part IV: A correct answer will receive 6 credits. Partial credit can be earned.
<BR><BR>
<BR><em>37. Jim had a bag of coins. The number of nickels, n, and the number of quarters, q, totaled 28 coins.
The combined value of the coins was $4.
<P>Write a system of equations that models this situation.
<P>Use your system of equations to algebraically determine both the number of quarters, q, and
the number of nickels, n, that Jim had in the bag.
<P>Jim was given an additional $3.00 that was made up of equal numbers of nickels and quarters.
How many of each coin was he given? Justify your answer.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Write one equation for the total number of the nickels and quarters, and then write a second equation for the total value of those nickels and quarters.
<CENTER>n + q = 28 <BR>5n + 25q = 400</CENTER>
<P>Use substition or elimination to solve the equation. For example, you could replace q with 28 - n.</P>
<CENTER> <P>5n + 25(28 - n) = 400</P>
<P>5n - 25n + 700 = 400</P>
<P>-20n = -300</P>
<P>n = 15</P>
<P>15 + q = 28</P><P>q = 13</P>
</CENTER>
<P>If he was given an equal amount of nickels and quarters, then n = q. Therefore, </P>
<CENTER><P>5q + 25q = 300 <BR>30q = 300<BR>q = 10<br>n = 10</P></CENTER>
<P>He received 10 more nickels and 10 more quarters.</P>
<BR><BR><BR>
End of Exam
<P>How did you do?</P>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-73409117718060965902024-02-15T14:31:00.000-05:002024-02-15T14:31:03.533-05:00January 2024 Geometry Regents Part III
<BR><P>This exam was adminstered in January 2024. </P>
<!-- <BR>The answers to Part I can be found <a href="https://mrburkemath.blogspot.com/2019/04/january-2019-common-core-geometry.html">here</a>
<BR>The answers to Parts III and IV can be found <a href="https://mrburkemath.blogspot.com/2019/03/january-2019-common-core-geometry.html">here</a> -->
<font size=4 face=helvetica>
<P><h3>January 2024 Geometry, Part III</h3>
<P>Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.
<BR><BR><BR><P><strong>32.</strong> <em>
Trish is a surveyor who was asked to estimate the distance across a pond. She stands at point C,
85 meters from point D, and locates points A and B on either side of the pond such that A, D, and
B are collinear.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3I8AcxTupSgO9JBUjCb4w6NvkZFkgYmlznXZ5z1UBmypTQjThrRVJ88GcB_4spYn1jlSdbLm2_rfgX3_0dbwpkPM0_jEyRzIyOjF0uIXjZn1IZJBKu3i9_0xKEsE7COR-U9_-TzNupYQClSm7BZKdnZpEOpr_oUlD3Q3Psry3ayp1IAtxMKUlwA/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="207" data-original-width="494" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3I8AcxTupSgO9JBUjCb4w6NvkZFkgYmlznXZ5z1UBmypTQjThrRVJ88GcB_4spYn1jlSdbLm2_rfgX3_0dbwpkPM0_jEyRzIyOjF0uIXjZn1IZJBKu3i9_0xKEsE7COR-U9_-TzNupYQClSm7BZKdnZpEOpr_oUlD3Q3Psry3ayp1IAtxMKUlwA/s1600/temp00.png"/></a></div>
<BR>Trish approximates the measure of angle DCB to be 35° and the measure of angle ACD to be 75°.
<BR>Determine and state the distance across the pond, AB, to the nearest meter.
</em>
<P><b>Answer: </b> </P>
<BR>Using the 75 degree angle, you can find the length of AD, which we'll call x. Using the 35 degree angle, you can find the length of DB, which we'll call y. Add the two together to find the length of AD.
<P>In both cases, we have the opposite side and we need to find the adjacent side. So we need to use the tangent ratio twice.</P>
<P>tan 75 = x / 85 <BR>x = 85 * tan 75 = 317.224... </P>
<P>tan 35 = y / 85 <BR>y = 85 * tan 35 = 59.517... </P>
<P>AB = x + y = 317.224 + 59.517 = 376.741 = 377 meters</P>
<BR><BR><BR><P><strong>33.</strong> <em> A candle in the shape of a right pyramid is modeled below. Each side of the square base measures 12 centimeters. The slant height of the pyramid measures 16 centimeters.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgb9J6Ea77exXUo0xC6uBthrsweWZvAZUaLSGaEt5rNgrlQWMIlD94tf1qj93qTu8hZFmgmwpCdbeU1qAWV-0L_nVIzrZ97ck9snG5PhqVQiadAOh0uyWQ2QzvdUaSCvXVQfNPJ8b8-l5x1TlK4rMy1ahdtHkUe24FJe-_4kEdPNzLFvVg1ilMJrA/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="235" data-original-width="261" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgb9J6Ea77exXUo0xC6uBthrsweWZvAZUaLSGaEt5rNgrlQWMIlD94tf1qj93qTu8hZFmgmwpCdbeU1qAWV-0L_nVIzrZ97ck9snG5PhqVQiadAOh0uyWQ2QzvdUaSCvXVQfNPJ8b8-l5x1TlK4rMy1ahdtHkUe24FJe-_4kEdPNzLFvVg1ilMJrA/s1600/temp00.png"/></a></div>
<BR> Determine and state the volume of the candle, to the nearest cubic centimeter
<BR>The wax used to make the candle weighs 0.032 ounce per cubic centimeter. Determine and state
the weight of the candle, to the nearest ounce.
</em>
<P><b>Answer: </b></P>
<BR>Notice that they gave you the <I>slant height</I> and not the height. You need the height to find the Volume. If you take a vertical slice (cross-section) of the pyramind, you would get an isosceles triangle with a base of 12 and two legs that were 16 cm. If you draw an altitude, you will get two congruent right triangles with a base of 6 and a hypotenuse of 16. Use this information to find the height.
<CENTER><P>(6)<sup>2</sup> + b<sup>2</sup> = (16)<sup>2</sup> <BR>
36 + b<sup>2</sup> = 256<BR>
b<sup>2</sup> = 220<br>
b = √(220) = 14.832...</P></CENTER>
<P>Use this value to find the Volume.</P>
<P>V = (1/3) Area of Base * height = (1/3) * 12 * 12 * 14.832 = 711.936 = 712 cu cm.</P>
<P>The weight is equal to the Volume times the Density: W = (712) * (0.032) = 22.784 = 23</P>
<BR><BR><BR><P><strong>34.</strong> <em> In the diagram of quadrilateral ABCD below, AB ≅ CD, and AB || CD.
<BR>Segments CE and AF are drawn to diagonal BD such that BE ≅ DF.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQdgIFJzcBhCe2QL4zcVHKphREecw3CmbcrhcBz_hj1qNIzndtHpED8BoYh3WnLrNfI0u6tKir3b1So4FlaqHhldefKwvnOZXG6VCfEircU02X3W8KUaGAxIr8eJ2X1kPkzhyphenhyphenMJmL6J5KRGquGIfh9mVDrBRSsw-SeJC-EndkA3wiF4-yX0yVAgA/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="129" data-original-width="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQdgIFJzcBhCe2QL4zcVHKphREecw3CmbcrhcBz_hj1qNIzndtHpED8BoYh3WnLrNfI0u6tKir3b1So4FlaqHhldefKwvnOZXG6VCfEircU02X3W8KUaGAxIr8eJ2X1kPkzhyphenhyphenMJmL6J5KRGquGIfh9mVDrBRSsw-SeJC-EndkA3wiF4-yX0yVAgA/s1600/temp00.png"/></a></div>
<BR>Prove: CE ≅ AF
<BR>
</em>
<P><b>Answer: </b></P>
<BR>TO prove that CE is congruent to AF, you are going to have to show that triangles BEC and DFA are congruent and then use CPCTC. To show that the triangles are congruent, you can use SAS.
<P>Your proof should look like this:</P>
<P>
<center>
<table width=90% border=5 cellspacing = 2 cellpadding = 2>
<tr>
<td width = 40%>Statement
</td>
<td>Reasons
</td>
</tr>
<tr>
<td>Quadrilateral ABCD, AB ≅ CD, AB || CD, and BE ≅ DF.
</td>
<td>Given
</td>
</tr>
<tr>
<td>ABCD is a parallelogram
</td>
<td>A quadrilateral with one pair of sides that are parallel and congruent is a parallelogram
</td>
</tr>
<tr>
<td>BC ≅ AD
</td>
<td>Opposite sides of parallelograms are congruent.
</td>
</tr>
<tr>
<td>BC || AD
</td>
<td>Opposite sides of parallelograms are parallel.
</td>
</tr>
<tr>
<td>∠ CBE ≅ ∠ ADF
</td>
<td>Alternate Interior Angles
</td>
</tr>
<tr>
<td>△BCE ≅ △DAF
</td>
<td>SAS Postulate
</td>
</tr>
<tr>
<td>(AB) / (AE) = (TR) / (TE)
</td>
<td>Corresponding sides of similar triangles are proportional
</td>
</tr>
<tr>
<td>CE ≅ AF
</td>
<td>CPCTC
</td>
</tr>
</table>
</center>
<P><B>End of Part III</b>
<P>How did you do?
<P>Questions, comments and corrections welcome.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-72443177829135193722024-02-13T12:17:00.000-05:002024-02-13T12:17:56.352-05:00January 2024 Algebra 1 Regents Part III<font size=4 face="Times New Roman">
<BR><P>This exam was adminstered in January 2024 . </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>January 2024 </h3></u></em>
<P>Part III: Each correct answer will receive 4 credits. Partial credit can be earned.
<BR><BR>
<BR><em>33. While playing golf, Laura hit her ball from the ground. The height, in feet, of her golf ball can be
modeled by h(t) = - 16t + 48t, where t is the time in seconds.
<BR>Graph h(t) on the set of axes below.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz8MsUS3Ev5at0TD7YyaZNV0i5MLDDUvVkrsybBsBmsVflZxVrVZU-4L8lQT0ta1kcsXSr0dxQS0ZnAaGjO6y9BOb3k1xBMtDKvhEk3UeRN69cKs_hPckA5iD4bIKoWpoYHciuVvWec7iSoWxfrI56QS-xLnC92g5-UQfqhKpVu0Js2XXvcLLCHQ/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="541" data-original-width="343" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz8MsUS3Ev5at0TD7YyaZNV0i5MLDDUvVkrsybBsBmsVflZxVrVZU-4L8lQT0ta1kcsXSr0dxQS0ZnAaGjO6y9BOb3k1xBMtDKvhEk3UeRN69cKs_hPckA5iD4bIKoWpoYHciuVvWec7iSoWxfrI56QS-xLnC92g5-UQfqhKpVu0Js2XXvcLLCHQ/s1600/temp00.png"/></a></div>
<BR>What is the maximum height, in feet, that the golf ball reaches on this hit?
<BR>How many seconds does it take the golf ball to hit the ground?
</em>
<P><B>Answer: </B> </P>
<BR>Look at the graph below. You can put the equation in the graphing calculator. You might want to set it so that it shows every 0.5 increment of x because the high point is going to happen at x = 1.5
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0CklalQFx6Vj9j9VLc7xm10Dzp1OwrO58erw0GF70XLW-Do_FjZR0AxZOaSyJrABBfHlRBSoYSCrOHoCHUYtIewZTITJi175gNs_uK0y6jNmW71uhwLS9s0USHa-RqiVA5vf98kQw4qYDMkA9nraf4e7no9E51F6EuP15EKxsQ8NPcGoBpsWyEw/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="541" data-original-width="343" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0CklalQFx6Vj9j9VLc7xm10Dzp1OwrO58erw0GF70XLW-Do_FjZR0AxZOaSyJrABBfHlRBSoYSCrOHoCHUYtIewZTITJi175gNs_uK0y6jNmW71uhwLS9s0USHa-RqiVA5vf98kQw4qYDMkA9nraf4e7no9E51F6EuP15EKxsQ8NPcGoBpsWyEw/s1600/temp00.png"/></a></div>
<P>The maximum height the ball reaches is 36 feet (which happens at 1.5 seconds).</P>
<P>It takes 3 seconds for the ball to hit the ground.
</P>
<BR><BR><BR>
<BR><em>34. The table below shows the number of SAT prep classes five students attended and the scores they
received on the test.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7FU0TJ08TfCIA1q-p5jiAXhEQ7Mc-Komnjo4y15A6jpm1oKcibrRvRloW7TvbYqPotXM82vs7GRBF7ayeG3ziAbMLBAC9QVGCpjI_ktRNTJEpRdtTom9jzr4ooskRAoODnbNWzmNbB9vFBEGpozT43E8j0L0LzWzsTn1zb-Hnxv4gQ0DKWlA23Q/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="129" data-original-width="551" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7FU0TJ08TfCIA1q-p5jiAXhEQ7Mc-Komnjo4y15A6jpm1oKcibrRvRloW7TvbYqPotXM82vs7GRBF7ayeG3ziAbMLBAC9QVGCpjI_ktRNTJEpRdtTom9jzr4ooskRAoODnbNWzmNbB9vFBEGpozT43E8j0L0LzWzsTn1zb-Hnxv4gQ0DKWlA23Q/s1600/temp00.png"/></a></div>
<BR>State the linear regression equation for this data set, rounding all values to the nearest hundredth.
<BR>State the correlation coefficient, rounded to the nearest hundredth.
<BR>State what this correlation coefficient indicates about the linear fit of the data.
</em>
<P>
<P><B>Answer: </B> </P>
<br>Enter all the data into two lists in your graphing calculator. You will have to run a linear regression. Make sure you have DIAGNOSTICS ON set on your calculator.
<P>When you run the linear regression, you will get a = 40.48 and b = 363.81, rounded to the nearest hundredth.
<BR>So the equation is y = 40.48x + 363.81</P>
<P>The correlation coefficient, r, is 0.84.</P>
<P>There is a strong positive correlation between the number of SAT prep courses attended and the score on the Math SAT.</P>
<BR><BR><BR>
<BR><em>35. Julia is 4 years older than twice Kelly’s age, x. The product of their ages is 96.
<BR>Write an equation that models this situation.
<BR>Determine Kelly’s age algebraically.
<BR>State the difference between Julia’s and Kelly’s ages, in years.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Kelly's age is <i>x</i>. Write an expression for Julia in terms of x. The product of that expression and x will be 96. Solve the quadratic equation that results from it.
<P>J = 2x + 4
<br>x(2x + 4) = 96</P>
<CENTER>
<P>2x<sup>2</sup> + 4x = 96
<BR>2x<sup>2</sup> + 4x - 96 = 0
<BR>x<sup>2</sup> + 2x - 48 = 0
<BR>(x + 8)(x - 6) = 0
<BR>x + 8 = 0 or x - 6 = 0
<BR>x = -8 or x = 6</P>
</CENTER>
<P>Throw out the negative answer because age cannot be negative. Therefore, Kelly is 6 years old.</P>
<P>
<P>Julia is 2(6) + 4 = 16. The difference between their ages is 10 years. </P>
<P>If you messed up the signs and thought that Kelly was 8 years old, then Julia would 20, and the difference would be 12 years. If you made one mistake, you would have lost only one point if the rest of your answers were consistent with that mistake.</P>
<BR><BR><BR>
<BR><em>36. On the set of axes below, graph the following system of inequalities:
<CENTER>2x - y > 4 <BR>x + 3y > 6</CENTER><BR>
Label the solution set S.
<P>Is (4,2) a solution to this system? Justify your answer.
<br>
</em>
<P>
<P><B>Answer: </B> </P>
<BR>Rewrite the inequalities into slope-intercept form. Remember when you divide an inequality by a negative number, you have to flip the direction of the inequality symbol.
<CENTER><P>2x - y > 4 <BR> - y > -2x + 4 <BR> y < 2x - 4</P>
<P>x + 3y > 6 <BR>3y > -x + 6 <BR>y > -1/3 x + 2</P>
</CENTER>
<P>Both inequalites will have broken lines. Shade above the line y > 1/3x + 2, and below y < 2x - 4. Mark the area with the crisscross with a big "S". This is your solution.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjGUeZaHjIs_aTtUFvSqfZwDjYNuLr6EF9rcVMx9Z6gHD8NVCJyTZFAzZbE7amH83dnlWBkE89hzAeloZL0PiZgToNhRhdPCLzMGYfUNE27PaKq94_TbR5pXzKvviUqlurwpaxXSo_jOWMiIU82IrhtRiNppJSi9oZggOh3KdlY9m-vb7VJKZdLQ/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="582" data-original-width="574" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjGUeZaHjIs_aTtUFvSqfZwDjYNuLr6EF9rcVMx9Z6gHD8NVCJyTZFAzZbE7amH83dnlWBkE89hzAeloZL0PiZgToNhRhdPCLzMGYfUNE27PaKq94_TbR5pXzKvviUqlurwpaxXSo_jOWMiIU82IrhtRiNppJSi9oZggOh3KdlY9m-vb7VJKZdLQ/s1600/temp00.png"/></a></div>
<P>(4,2) is a solution to the system because it's in the double-shaded area.</P>
<BR><BR><BR>
End of Part III
<P>How did you do?</P>
<BR><BR><BR>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-15680721096247609702024-02-12T13:10:00.001-05:002024-02-12T13:10:54.467-05:00A Fine Line<font size="2">(Click on the comic if you can't see the full image.)</font><br />
<center>
<aa href="http://xwhy.comicgenesis.com/d/20240212.html">
<iii src="http://mrburkemath.net/xwhy/images/1776 .png" title="When you give all you have to give, that's giving 100%. The math checks. "></iii>
</aa>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIYaflVOoOuW0hLrboExmNxAZnPDSdwwODolvZcr2ugnpCxLNiHDJJa6gfuKtABAjU6i9LijmiNyiMMykyIxUpI-m8FNGTb0t4DKU0by83Se16ySLQTvHJ2-RWeqdvi5eUZu88VTTp2_lyDP4TFbEuvqYfGJBgCVAudn3kR1Hl-XVL24SyWSPAXw/s1600/fineline01.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="350" data-original-width="750" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIYaflVOoOuW0hLrboExmNxAZnPDSdwwODolvZcr2ugnpCxLNiHDJJa6gfuKtABAjU6i9LijmiNyiMMykyIxUpI-m8FNGTb0t4DKU0by83Se16ySLQTvHJ2-RWeqdvi5eUZu88VTTp2_lyDP4TFbEuvqYfGJBgCVAudn3kR1Hl-XVL24SyWSPAXw/s1600/fineline01.png"/></a></div>
<font color="FF5544">(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).</font>
</center>
<font size=4>
<P>There's a fine line between humor and informational context.</P>
<P>You know there's a great chance that Mike is not only aware that it was a joke but it's something that he would've said.</P>
<P>And everyone likes Mike. Right? And I ask this not because people say he's my avatar in this comic.</P>
<P>I almost had Scott tell it to Ken, but their rivalry is a little different (bordering, perhaps, on animosity). </P>
</font>
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
<br /><br /><i>Come back often for more funny math and geeky comics.</i>
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<br />(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-32281067506918423062024-02-10T15:00:00.002-05:002024-02-19T10:26:18.607-05:00January 2024 Geometry Regents Part II
<BR><P>This exam was adminstered in January 2024 . </P>
<!-- <BR>The answers to Part I can be found <a href="https://mrburkemath.blogspot.com/2019/04/january-2019-common-core-geometry.html">here</a>
<BR>The answers to Parts III and IV can be found <a href="https://mrburkemath.blogspot.com/2019/03/january-2019-common-core-geometry.html">here</a> -->
<font size=4 face=helvetica>
<P><h3>January 2024 Geometry, Part II</h3>
<P>Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.
<BR><BR><BR><P><strong>25.</strong> <em>
In isosceles triangle ABC shown below, AB ≅ AC, and altitude AD is drawn.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKci_eGQrXfZUw6AqlQTb7J1x477DGJibMHh-dkJcrckzv6b-qBEh5fzlDvY31FIKAhT-CJnbSvtRdZlzMtSSIn9f78uUVEw_Fp7F_ZYBuOmGhTeqEyL_1SMrdBCxuVHaq0b5xvSv4lR4W5GwYLVbXmf_QpDpQ_jMqZ_hfC5ER7PMBL9HT1XMOtA/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="189" data-original-width="216" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKci_eGQrXfZUw6AqlQTb7J1x477DGJibMHh-dkJcrckzv6b-qBEh5fzlDvY31FIKAhT-CJnbSvtRdZlzMtSSIn9f78uUVEw_Fp7F_ZYBuOmGhTeqEyL_1SMrdBCxuVHaq0b5xvSv4lR4W5GwYLVbXmf_QpDpQ_jMqZ_hfC5ER7PMBL9HT1XMOtA/s1600/temp00.png"/></a></div>
<P>The length of AD is 12 cm and the length of BC is 10 cm.
<BR>Determine and state, to the nearest cubic centimeter, the volume of the solid formed by
continuously rotating triangle ABC about AD.
</em>
<P><b>Answer: </b> </P>
<BR>Rotating triangle ABC around AD would create a cone with a height equal to AD (which is 12 cm) and a radius equal to BD (which is half of 10, or 5).
<P>Use the formula for the Volume of a cylinder.
</P>
<P>V = (1/3) π r<sup>2</sup> h = (1/3) π (5)<sup>2</sup> (12) = 314.159 ... = 314</P>
<P>Amusingly, 12 * 25 = 300, and 1/3 of 300 is 100, so the answer is 100 pi, rounded.</P>
<P>(Okay, that might only be amusing to me, but I'm amused and you may choose to be as well.</P>
<BR><BR><BR><P><strong>26.</strong> <em> The diagram below models the projection of light from a lighthouse, L. The sector has a
radius of 38 miles and spans 102°.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWBOYaT0YiKx1tMghd9o4pi__9xWOSyrtMKvfgj_h8vAsilbyI3LXVtOcDXjx-wRVCDYwKFnhZPsLys_a-lJ8f_EidyqML2AOQ9w-avUIfDtqMuOiU45gYsOBTq5JMgfnHXY4DlQ3jRjfYOLU-P4v9DGjPjcCLRMYuFxESLj8lehGMmWghfhe7nw/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="176" data-original-width="275" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWBOYaT0YiKx1tMghd9o4pi__9xWOSyrtMKvfgj_h8vAsilbyI3LXVtOcDXjx-wRVCDYwKFnhZPsLys_a-lJ8f_EidyqML2AOQ9w-avUIfDtqMuOiU45gYsOBTq5JMgfnHXY4DlQ3jRjfYOLU-P4v9DGjPjcCLRMYuFxESLj8lehGMmWghfhe7nw/s1600/temp00.png"/></a></div>
<BR>Determine and state the area of the sector, to the nearest square mile.
<P>PLEASE, PLEASE, PLEASE!!! Make sure you round correctly. It hurts to see students do all the work to answer this question and then lose half of the points because they didn't round correctly.</P>
</em>
<P><b>Answer: </b></P>
<BR>You find the area of a sector in the same way that you find the area of an entire circle. However, in the same way that a cone is only 1/3 of a cylinder of the same dimensions, a sector is only a fraction of the entire circle, and that fraction will be the central angle divided by 360 degrees.
<P>Use the formula for the Area of a Circle and multiply by the fraction.
</P>
<P>A = (102/360) π r<sup>2</sup> = (102/360) π (38)<sup>2</sup> = 1285.33... = 1285 square miles. </P>
<BR><BR><BR><P><strong>27.</strong> <em>A Segment CA is drawn below. Using a compass and straightedge, construct
isosceles right triangle CAT where CA is perpendicular to CT and CA ≅ CT.
[Leave all construction marks.]
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmjUhUli1raGonFVN7tHGFpTHK0pQrZ4U-xHb5plBCH2W2nYO1P9GtnEKiwW9_F4QjeBbuskA63iLXPM3eRunbb0vey0w9ex2NBSjyybo_SMd_geezt93S2tdD8QLNCQtPbm-nhKiTeH328K4GsBiTctbDLvXj1NqWOhupqiytza4LbKqG_FYwdQ/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="276" data-original-width="375" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmjUhUli1raGonFVN7tHGFpTHK0pQrZ4U-xHb5plBCH2W2nYO1P9GtnEKiwW9_F4QjeBbuskA63iLXPM3eRunbb0vey0w9ex2NBSjyybo_SMd_geezt93S2tdD8QLNCQtPbm-nhKiTeH328K4GsBiTctbDLvXj1NqWOhupqiytza4LbKqG_FYwdQ/s1600/temp00.png"/></a></div>
</em>
<P><b>Answer: </b></P>
<BR>Extend line CA past point C (to the left). Draw a semicircle with radius AC, mark the new point on line AC and call it point D.
<P>Construct a perpendicular bisector of DA through point C. Put the compass and point D and open it wider than the length of DC. Make arcs above and below line DA. Move the compass to point A without changing the width of the compass. Make marks above and below DA that intersect the arcs you just make.</P>
<P>Draw the vertical line between the two intersections and through point C.</P>
<P>Label one of the intersections point T. Use a straightedge to draw CT and TA. </P>
<P>You have constructed isosceles triangle CAT. Done.</P>
<P><div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj322WoLZR-xCtoehgbnF1AJMlFoTqivaWA4NH5mUx57SADlpwZVHTPywsufzKBz1d1dQT4LZCcAvw8Y7l7HVjkQoqpuK2UxryBTiJ0cIeZvalu9tJqtBDxotYeSHqX4vm1o2cAtO2jWGt11L8rGZ0-nVPq9hybYMprJqHa25IPt1v6SS72LnrJxg/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="443" data-original-width="468" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj322WoLZR-xCtoehgbnF1AJMlFoTqivaWA4NH5mUx57SADlpwZVHTPywsufzKBz1d1dQT4LZCcAvw8Y7l7HVjkQoqpuK2UxryBTiJ0cIeZvalu9tJqtBDxotYeSHqX4vm1o2cAtO2jWGt11L8rGZ0-nVPq9hybYMprJqHa25IPt1v6SS72LnrJxg/s1600/temp00.png"/></a></div>
</P>
<BR><BR><BR><P><strong>28.</strong> <em>A On the set of axes below, congruent triangles ABC and DEF are graphed.
<P>Describe a sequence of rigid motions that maps triangle ABC onto triangle DEF.
</em>
<P><b>Answer: </b></P>
<BR>This is a "mean" question for two reasons. First, there is no reason to write a "sequence" because it can be done in one rotation. Second, it's set up in such a way to encourage a student to map ABC onto FED instead of DEF. Yes, that matters.
<b> <P>A rotation of 90 degrees counterclockwise around the origin would map ABC onto DEF.</P></b>
<P>A reflection over the y-axis and a translation of +1,+1 would map to the FED and receive half credit.</P>
<P>Under the new curriculum in NYS, you could also state, "Translate triangle ABC along ray AD until A comes to D. Then rotate ABC until point B coincides with point E."
<BR>(I'm not kidding.)</P>
<BR><BR><BR><P><strong>29.</strong> <em>In triangle ADC below, EB is drawn such that AB = 4.1, AE = 5.6, BC = 8.22, and ED = 3.42.
<BR> Is triangle ABE similar to trianlge ADC? Explain why.</em>
<P><b>Answer: </b>
<BR>There's a little bit of a hint when they didn't state "or why not".
<P>Set up a proportion of corresponding sides. Remember to add the lines segments to get the lengths of AD and AC. We don't know anything about the lengths of BE or CD, but we do know that both triangles share angle A, which, of course, is congruent to itself because of the Reflexive Property.
<P> Is 4.1 / (5.6 + 3.42) = 5.6 / (4.1 + 8.22) ? </P>
<P>Dividing, we get 0.4545... = 0.4545...</P>
<P>Or by cross-multiplying:</P>
<P>(5.6 + 3.42) (5.6) = 50.512; (4.1) (4.1 + 8.22) = 50.512</P>
<P>Since the corresponding sides are proportional, and because the two triangles share angle A, the triangles are similar by SAS.</P>
<P>If you didn't show the sides were propotional or you didn't mention angle A and SAS, you couldn't receive full credit.</P>
<BR><BR><BR><P><strong>30.</strong> <em>
Determine and state the coordinates of the center and the length of the radius of the circle
represented by the equation x<sup>2</sup> + 16x + y<sup>2</sup> + 12y - 44 = 0.
</em>
<P><b>Answer: </b>
<BR>You have to complete the squares for both the x terms and the y terms to get the equation into the form (x - h)<sup>2</sup> + (y - k)<sup>2</sup> = r<sup>2</sup>, where (h,k) is the center of the circle and r is the radius of the circle. (Not r<sup>2</sup> -- don't make that mistake!)
<P>Has of 16 is 8, and 8 squared is 64. Half of 12 is 6, and 6 squared is 36. Add 64 and 36 to both sides of the equation. This means that (x + 8) and (y + 6) will both be in the final answer.</P>
<CENTER>
<P>x<sup>2</sup> + 16x + y<sup>2</sup> + 12y - 44 = 0</P>
<P>x<sup>2</sup> + 16x + 64 + y<sup>2</sup> + 12y + 36 - 44 = 64 + 36</P>
<P>x<sup>2</sup> + 16x + 64 + y<sup>2</sup> + 12y + 36 = 100 + 44</P>
<P>(x + 8)<sup>2</sup> + (y + 6)<sup>2</sup> = 144</P>
<P>(x + 8)<sup>2</sup> + (y + 6)<sup>2</sup> = 12<sup>2</sup></P>
</CENTER>
<P>The center of the circle is (-8,-6), and the radius is 12.</P>
<P>Both forget to "flip the signs" of the coordinates and to take the square root to find the radius.
</P>
<BR><BR><BR><P><strong>31.</strong> <em> In the diagram below, traingle SBC ~ triangle CMJ and cos J = 3/5.
<BR>Determine and state m∠S, to the nearest degree.
</em>
<P><b>Answer: </b>
<BR>This is another question that I do not like BECAUSE they hide important information. Since they stated that △SBC ~ △CMJ, then ∠S corresponds to ∠C, not ∠J.
<P>This is necessary information. The problem is that in my years of teaching, many educators are a little lax in their naming conventions when stating the order of the vertices in the triangles. This could confuse many students. What's more since triangle SBC is the same as triangle BCS and triangle CBS, it isn't WRONG, per se, to write the letters in the incorrect order because it's still true.
</P>
<P>So the entire question hinges on convention and whether or not the instructor followed it.</P>
<P>If you know enough about right triangles, you know enough to remember that if cos J = 3/5, then sin J = 4/5, and sin JCM = 3/5 and cos JCM = 4/5, because it is a multiple of a 3-4-5 triangle.</P>
<P>You can use trig ratios to find angles or Pythogean Theorem but that's a simple fact. </P>
<P>Since angle S correspond to angle C, then sin S = 3/5, and S = sin <sup>-1</sup> (3/5) = 37 degrees.</P>
<P><B>End of Part II</B>
<P>How did you do?
<P>Questions, comments and corrections welcome.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-48592484517308965882024-02-10T11:50:00.000-05:002024-02-10T11:54:47.025-05:00January 2024 Algebra 1 Regents Part II<font size=4 face="Times New Roman">
<BR><BR>This exam was adminstered in January 2024. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>January 2024</h3></u></em>
<P>Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>25. Student scores on a recent test are shown in the table below.
<CENTER><TABLE align=center width=50% border=2 cellspacing=2 cellpadding=2>
<TR align=center><td>
85</td><td> 96</td><td> 92</td><td> 82</td><td> 90</td></TR>
<TR align=center><td> 90</td><td> 88</td><td> 95</td><td> 85</td><td> 88</td></TR>
<TR align=center><td> 90</td><td> 87</td><td> 96</td><td> 82</td><td> 85</td></TR>
<TR align=center><td> 92</td><td> 96</td><td> 85</td><td> 92</td><td> 87</td></TR>
</TABLE>
</CENTER>
<P>On the number line below, create a dot plot to model the data</P>
<P>State the median test score for the data set.</P>
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>For each number in the table place a dot about that value on the number line. Make them all the same size so that, say, 3 dots above one number are the same height as 3 dots above another. Don't make any boxes or bars. You should have 20 dots when you are finished. If you have a different number of dots, you either left something out or repeated some piece of data.
<P>Note that having exactly 20 dots is not a guarantee that you put them in the right place, but having 19 or 21 is definitely an error.</P>
<P>Your graph should look like this one:</P>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTDc-8stw7GDlJ7Y9sbSGtxuFhFaJJEFxf4ZCCimXyBFkwfV1xmYK3x8q61BMRStFN83rh8ebkdRy6Wx8h9MkSVCMhDzjWqUQ-dP79G5o4LYUy2_NNRHtPA8R9HPrESdF0uyvuTqsRJbHxhiYugBfgNW4ETAPAemxtECniOcEGzh3j6qyuL2wILQ/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="178" data-original-width="534" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTDc-8stw7GDlJ7Y9sbSGtxuFhFaJJEFxf4ZCCimXyBFkwfV1xmYK3x8q61BMRStFN83rh8ebkdRy6Wx8h9MkSVCMhDzjWqUQ-dP79G5o4LYUy2_NNRHtPA8R9HPrESdF0uyvuTqsRJbHxhiYugBfgNW4ETAPAemxtECniOcEGzh3j6qyuL2wILQ/s1600/temp00.png"/></a></div>
<P>There are 20 pieces of data, so the median test score will be the average of the 10th and the 11th. The 10th is 88 and the 11th is 90. The number in the middle of 88 and 90 is 89, which is the median.</P>
<P>
<BR><BR><BR>
<BR><em>26. State whether 2√(3) + 6 is rational or irrational. Explain your answer.
</em>
<P>
<P><B>Answer: </B> </P>
<br>2√(3) is an irrational number and the sum of a rational and an irrational number is always irrational.
<P>If you mentioned that the decimal value goes on forever you must mention that it doesn't have a repeating pattern.</P>
<BR><BR><BR>
<BR><em>27. The table below shows data from a recent car trip for the Burke family.
<CENTER><TABLE align=center width=60% border=2 cellspacing=2 cellpadding=2>
<TR align=center><td>
<b>Hours After Leaving </b>(x)</td><td> 1</td><td> 2</td><td> 3</td><td> 4</td><td> 5</td></TR>
<TR align=center><td>
<b>Miles from Home </b>(y)</td><td> 45</td><td> 112</td><td> 178</td><td> 238</td><td> 305</td></TR>
</TABLE>
</CENTER>
<P>State the average rate of change for the distance traveled between hours 2 and 4.
Include appropriate units.</P>
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Divide the difference of the miles at hour 4 and miles at hour 2 by the difference of 4 minus 2.
<P>(238 - 112) / (4 - 2) = 63 mph.</P>
<P>It says to add appropriate units, so if you don't specify mph or miles per hour, you will lose a point.</P>
<BR><BR><BR>
<BR><em>28. On the set of axes below, graph the equation 3y + 2x = 15.
<P>Explain why (-6,9) is a solution to the equation.</P>
</em>
<P>
<P><B>Answer: </B> </P>
<BR>Rewrite the equation into slope-intercept form to graph (or to put in your graphing calculator).
<P><CENTER>3y + 2x = 15 <BR> 3y = -2x + 15 <BR> y = -2/3 x + 5</CENTER></P>
<P>The slope is -2/3 and the y-intercept is 5. Start at (0,5) and use the rise (-2) and run (3) to find points on the line. Or use the table of values in the calclulator.</P>
<P>Your graph will look like this:</P>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhendiUKLjM7GeN1tn1P2zEUFVSvaR3WSBh7ns8uE8QCTjXEtIa-BXMLz-hGLwc0xynzBvvUxZotoGwKPnKqV1k0WJgNliZ7bT3cSQzzKAUxzYpgEuw_doCKLA6oHkUeEG9dV-D2Jdho91EKZkENI-HySHo23JLE645KP2NezNtCYloLRNRZJe2-Q/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="579" data-original-width="605" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhendiUKLjM7GeN1tn1P2zEUFVSvaR3WSBh7ns8uE8QCTjXEtIa-BXMLz-hGLwc0xynzBvvUxZotoGwKPnKqV1k0WJgNliZ7bT3cSQzzKAUxzYpgEuw_doCKLA6oHkUeEG9dV-D2Jdho91EKZkENI-HySHo23JLE645KP2NezNtCYloLRNRZJe2-Q/s1600/temp00.png"/></a></div>
<P>The point (-6,9) is a solution to the equation because it is a point on the line. All points on the line are solutions to the equation.</P>
<P>
<BR><BR><BR>
<BR><em>29.Using the quadratic formula, solve 3x<sup>2</sup> - 2x - 6 = 0 for all values of x.
<BR>Round your answers to the nearest hundredth.
</em>
<P>
<P><B>Answer: </B> </P>
<BR>Plug the values into the quadratic formula and evaluate. Don't forget to find two solutions.
<P>Use a = 3, b = -2, and c = -6.</P>
<CENTER>
<P> x = ( -b <u>+</u> √( (b)<sup>2</sup> - 4(a)(c) ) / ( 2a )</P>
<P> x = ( -(-2) <u>+</u> √( (-2)<sup>2</sup> - 4(3)(-6) ) / ( 2(3) )</P>
<P> x = ( 2 <u>+</u> √( 4 + 72) ) / (6)</P>
<P> x = ( 2 + √( 76) ) / (6) or x = ( 2 - √( 76) ) / (6)</P>
<P>x = 1.7862... or x = -1.1196...</P>
</CENTER>
<P>x = 1.79 or x = -1.12</P>
<BR><BR><BR>
<BR><em>30. The piecewise function f(x) is given below.
<CENTER> f(x) = { 2x - 3, x > 3; <BR>-x<sup>2</sup> + 15, x <u><</u> 3 }
</CENTER>
<BR>State the value of f(3).
<BR>Justify your answer.
</em>
<P>
<P><B>Answer: </B> </P>
<BR>Since 3 is not great than 3, but three is less than or equal to 3, use the second piece of the function.
<P>f(3) = -(3)<sup>2</sup> + 15 = -9 + 15 = 6</P>
<P>If you evaluate the top piece instead, you will get 1 credit. </P>
<BR><BR><BR>
<BR><em>31. Express the equation x<sup>2</sup> - 8x = -41 in the form (x - p)<sup>2</sup> = q.
</em>
<BR>
<P><B>Answer: </B> </P>
<BR>Complete the square by adding 16 to both sides. Half of -8 is -4 and (-4) square is 16.
<CENTER>
<P>x<sup>2</sup> - 8x = -41<br>x<sup>2</sup> - 8x + 16 = -41 + 16<BR>(x - 4)<sup>2</sup> = -25
</P></CENTER>
<P>
<BR><BR><BR>
<BR><em>32. Factor 36 - 4x<sup>2</sup> completely.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Remember that when it says "completely", there is usually more than one step. In this case there is a common factor of 4 in the two terms.
<CENTER>
<P>36 - 4x<sup>2</sup> <BR>4(9 - x<sup>2</sup>)<BR>4(3 - x)(3 + x)</P>
</CENTER>
<P>The difference of two perfect squares always factors into two conjugates. (That is, two binomials with the same two terms, except that one has a + and the other has a -.)</P>
<P>
<BR><BR><BR>
End of Part II
<P>How did you do?</P>
<BR><BR><BR>
More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-25915551475131737342024-02-05T18:35:00.004-05:002024-02-05T18:35:48.668-05:00August 2023 Algebra 2 Regents Part IV<font size=4 face="Times New Roman">
<BR><P>This exam was adminstered in August 2023. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>Algebra 2 August 2023</h3></u></em>
<P>Part IV: A correct answer will receive 6 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>37.The Manford family started savings accounts for their twins, Abby and Brett, on the day they were
born. They invested $8000 in an account for each child. Abby’s account pays 4.2% annual interest
compounded quarterly. Brett’s account pays 3.9% annual interest compounded continuously.
Write a function, A(t), for Abby’s account and a function, B(t), for Brett’s account that calculates
the value of each account after t years.
<P>Determine who will have more money in their account when the twins turn 18 years old, and find
the difference in the amounts in the accounts to the nearest cent.
</P>
<P>Algebraically determine, to the nearest tenth of a year, how long it takes for Brett’s account to
triple in value.
</P>
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR><I>A</I> is an exponential growth function. <I>B</I> is continuously compounded and uses <I>e</I> instead.
<P><CENTER>A(t) = 8000(1 + .042/4)<sup>4t</sup> <BR> B(t) = 8000e<sup>.039t</sup></CENTER></P>
<P>For the second part, evaluate both functions for t = 18:</P>
<CENTER><P>A(18) = 8000(1 + .042/4)<sup>4(18)</sup> = 16970.899 = $16,970.90</P>
<P>B(18) = 8000e<sup>.039(18)</sup> = 16142.273 = $16,142.27</P></CENTER>
<P>The difference in the accounts will be 16970.90 - 16142.27 = $828.63.</P>
<P>To find when Brett's will triple in value, we need to find when B(t) = 24000.</P>
<CENTER><P>8000e<sup>.039t</sup> = 24000<BR>
e<sup>.039t</sup> = 24000/8000<BR>
ln(e<sup>.039t</sup>) = ln(24000/8000)<BR>
.039t = 1.09861...<BR>
t = 28.1695...</P></CENTER>
<P>In 28.2 years the value in Brett's account will triple.</P>
<P>
<BR><BR><BR>
End of Exam
<P>How did you do?</P>
<BR><BR><BR>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-70939897351704511992024-02-05T02:00:00.001-05:002024-02-05T02:00:00.253-05:00August 2023 Algebra 2 Regents, Part III<font size=4 face="Times New Roman">
<BR><P>This exam was adminstered in August 2023. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>Algebra 2 August 2023</h3></u></em>
<P>Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>33. 3 Sketch p(x) = -log<sub>2</sub>(x + 3) + 2 on the axes below.
<BR>Describe the end behavior of p(x) as x → -3.
<BR>Describe the end behavior of p(x) as x → ∞.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Put the equation into your graphing calculator and look at the table of values. If you release that log<sub>2</sub>(1) is equal to 0, then you can deduce that p(-2) = 0 + 2, so (-2,2) is a point on the graph.
<P>If you plot the points: (-2,2), (-1,1), (0,0.415), (1,0), (2,-0.322), and (3,-0.585), you'll have enough for a good sketch. There will be a vertical asymptote at x = -3.</P>
<P>Your graph should look like the following:</P>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLsAT4GW4OGflpa8IZZuK75LKdkNPPZwDtcjVKHbQ9kd7ZFXWY2nZfipX1ZrWW0zIFiIw09Na7HWRVMiEuBlSMzM26dlCQi44na3o-MGJjLA2n2gUjxp2dnbwhgtVMO1J7B5ZVmTad6XRSzoKGs7glp63F2dSrIFeayoQCsuCqr_NlKMxOEVtK7w/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="452" data-original-width="497" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLsAT4GW4OGflpa8IZZuK75LKdkNPPZwDtcjVKHbQ9kd7ZFXWY2nZfipX1ZrWW0zIFiIw09Na7HWRVMiEuBlSMzM26dlCQi44na3o-MGJjLA2n2gUjxp2dnbwhgtVMO1J7B5ZVmTad6XRSzoKGs7glp63F2dSrIFeayoQCsuCqr_NlKMxOEVtK7w/s1600/temp00.png"/></a></div>
<P>As x goes to -3, p(x) goes to infinity. As x goes to infinity, p(x) goes to negative infinity.</P>
<P>
<BR><BR><BR>
<BR><em>34. Solve for x algebraically: 1 / (x - 6) + x / (x - 2) = 4 / (x<sup>2</sup> - 8x + 12)
</em>
<P>
<P><B>Answer: </B> </P>
<br>You need to combine the fractions on the left if you wish to cross-multiply. Or you can multiply both sides of the equation by (x<sup>2</sup> - 8x + 12) if you realize that the polynomial factors into (x - 6)(x - 2).
<CENTER><P>1 / (x - 6) + x / (x - 2) = 4 / (x<sup>2</sup> - 8x + 12)<BR>
(x - 6)(x - 2) ( 1 / (x - 6) + x / (x - 2) ) = (4 / (x<sup>2</sup> - 8x + 12)) (x - 6)(x - 2)
<BR> x - 2 + x(x - 6) = 4
<BR>x - 2 + x<sup>2</sup> - 6x = 4
<BR>x<sup>2</sup> - 5x - 6 = 0
<BR>(x - 6)(x + 1) = 0
<BR>x - 6 = 0 or x + 1 = 0
<BR>x = 6 or x = -1</P>
</CENTER>
<P>Throw out x = 6 as extraneous because 1 / (6 - 6) is undefined.</P>
<P>The only solution is x = -1.</P>
<BR><BR><BR>
<BR><em>35. Solve the following system of equations algebraically for x, y, and z.
<BR><CENTER>
2x + 4y - 3z = 12 <BR>3x - 2y + 2z = -9 <BR> -x + y - 3z = 0
</CENTER>
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>You can rewrite the third equation to solve for either x or y and then substitute the expression into the first two equations. Then you can solve that system.
<P><CENTER>
-x + y - 3z = 0 <BR>y - 3z = x
</CENTER>
</P>
<CENTER>
<P> 2(y - 3z) + 4y - 3z = 12 <BR>3(y - 3z) - 2y + 2z = -9 </P>
<P> 2y - 6z + 4y - 3z = 12 <BR>3y - 9z - 2y + 2z = -9 </P>
<P> 6y - 9z = 12 <BR>y - 7z = -9 </P>
<P> 6y - 9z = 12 <BR>y = 7z - 9 </P>
<P> 6(7z - 9) - 9z = 12 <BR>
42z - 54 - 9z = 12 <BR>
33z = 66<BR>
z = 2 </P>
<P>y = 7(2) - 9 = 5</P>
<P>x = y - 3z = (5) - 3(2) = -1</P></CENTER>
<BR><BR><BR>
<BR><em>36. Two classes of students were entered into an experiment to see whether using an interactive
whiteboard leads to better grades. It was observed that the mean grade of students in the class
with the interactive whiteboard was 0.6 points higher than the class without it. To determine if the
observed difference is statistically significant, the classes were rerandomized 5000 times to study
these random differences in the mean grades. The output of the simulation is summarized in the
histogram below.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWi7trcm1k3t2TKj7fiHvvQteCHIhlEmmAXfmt58YEd63DyaSJhloqxuNQ3ozbxAsV8TYuBBF2uPnzucbbpkEalbZml3c7k3eDzwoAn_YR6Rg0ZpMb4YWI8WhJYnk2YyLDa7mRPROrwxmdVBWRxn79D4E8FaN2QlhQgnPaMpLIhrRESKP38eBiAg/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="250" data-original-width="477" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWi7trcm1k3t2TKj7fiHvvQteCHIhlEmmAXfmt58YEd63DyaSJhloqxuNQ3ozbxAsV8TYuBBF2uPnzucbbpkEalbZml3c7k3eDzwoAn_YR6Rg0ZpMb4YWI8WhJYnk2YyLDa7mRPROrwxmdVBWRxn79D4E8FaN2QlhQgnPaMpLIhrRESKP38eBiAg/s1600/temp00.png"/></a></div>
<BR>Determine an interval containing the middle 95% of the simulation results. Round your answer to
the nearest hundredth.
<BR>Does the interval indicate that the difference between the classes’ grades is significant? Explain.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>The interval is the mean plus or minus twice the standard deviation.
<P>0.01 + 2(0.38) = 0.77 <BR>0.01 - 2(0.38) = -0.75</P>
<P>The interval is [-0.75, 0.77]</P>
<P>Since 0.6 is within the interval, so it's not significant.</P>
<P>
<BR><BR><BR>
End of Part III
<P>How did you do?</P>
<BR><BR><BR>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-15443635202012759492024-02-04T13:34:00.001-05:002024-02-04T13:34:17.717-05:00August 2023 Algebra 2, Part II<font size=4 face="Times New Roman">
<BR>
<P>This exam was adminstered in August 2023. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>Algebra 2 August 2023</h3></u></em>
<P>Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>25. Factor the expression 2x<sup>3</sup> - 3x<sup>2</sup> - 18x + 27 completely.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Factor by grouping, and then factor the quadratic you get after the first step.
<P>There are two ways to group, and either should work in any question of this kind.</P>
<CENTER>
<P>2x<sup>3</sup> - 3x<sup>2</sup> - 18x + 27
<br>(2x<sup>3</sup> - 3x<sup>2</sup>) - (18x - 27)
<br>x<sup>2</sup>(2x - 3) - 9(2x - 3)
<br>(x<sup>2</sup> - 9)(2x - 3)
<br>(x + 3)(x - 3)(2x - 3)
</P></CENTER>
<P>You can also switch the two middle terms around. This is just the way I learned it, so I usually do it, especially if it helps me avoid factoring out a minus sign.</P>
<CENTER>
<P>2x<sup>3</sup> - 18x - 3x<sup>2</sup> + 27
<br>(2x<sup>3</sup> - 18x)(3x<sup>2</sup> - 27)
<br>2x(x<sup>2</sup> - 9) - 3(x<sup>2</sup> - 9)
<br>(2x - 3)(x<sup>2</sup> - 9)
<br>(2x - 3)(x + 3)(x - 3)
</P></CENTER>
<P>
<BR><BR><BR>
<BR><em>26. Algebraically determine the values of x that satisfy the system of equations shown below:
<CENTER>y = x<sup>2</sup> + 8x - 5 <BR>y = 8x - 4</CENTER>
</em>
<P>
<P><B>Answer: </B> </P>
<br>Set the two statements equal to each other. Solve the quadratic equation for x. You don't need to find y.
<CENTER><P>y = x<sup>2</sup> + 8x - 5 <BR>y = 8x - 4<BR>
<BR>x<sup>2</sup> + 8x - 5 = 8x - 4
<BR>x<sup>2</sup> - 1 = 0
<BR>(x - 1)(x + 1) = 0
<BR>x - 1 = 0 or x + 1 = 0
<BR>x = 1 or x = -1
</P></CENTER>
<BR><BR><BR>
<BR><em>27. Solve the equation 3x<sup>2</sup> + 5x + 8 = 0. Write your solution in a + bi form.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>When they tell you "a + bi" form, you know that it won't be something that you can factor easily and that you need to use the quadratic formula.
<CENTER>
<P>3x<sup>2</sup> + 5x + 8 = 0
</P>
<P>x = (-b <u>+</u> √(b<sup>2</sup> - 4ac) ) / (2a)
</P>
<P>x = (-5 <u>+</u> √(5<sup>2</sup> - 4(3)(8)) ) / (2(3))
</P>
<P>x = (-5 <u>+</u> √(25 - 96) ) / (6))
</P>
<P>x = (-5 <u>+</u> √(-71) ) / (6))
</P>
<P>x = -5/6 <u>+</u> <i>i</i> √(71)/6
</P>
</CENTER>
<BR><BR><BR>
<BR><em>28. On the coordinate plane below, sketch at least one cycle of a cosine function with a midline at y = -2, an amplitude of 3, and a period of π/2.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Cosine doesn't start at the midline. Cosine starts at the midline plus the amplitude, which in this case is -2 + 3 = 1. The minumim will occur at y = -2 - 3 = -5 when x = (1/2)(π/2) = π/4.
<P>Your sketch should look like this:</P>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzPDvhSZceb0X-8wTLA4Vcjz8EQ1aEuT0qWhaRab-KWOSLYYfMhXIvp_lIYu8vTVx4GMU3I4Z5HhlmYH9uH30vVI8narWq7TMBuivLeygM1XaUAJc9d6qFkczBLBRCt_6DcIdiU3_BYYr_Ivs9ZjVIop8297r322PERxKH95uKwFi0Y98pmFSh8g/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="314" data-original-width="386" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzPDvhSZceb0X-8wTLA4Vcjz8EQ1aEuT0qWhaRab-KWOSLYYfMhXIvp_lIYu8vTVx4GMU3I4Z5HhlmYH9uH30vVI8narWq7TMBuivLeygM1XaUAJc9d6qFkczBLBRCt_6DcIdiU3_BYYr_Ivs9ZjVIop8297r322PERxKH95uKwFi0Y98pmFSh8g/s1600/temp00.png"/></a></div>
<P>Remember that a sketch doesn't have to be perfect, but it shouldn't have any obvious errors, such as crossing the axis at the wrong place.
<P>
<BR><BR><BR>
<BR><em>29. Given i is the imaginary unit, simplify (5xi<sup>3</sup> - 4i)<sup>2</sup> as a polynomial in standard form.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Use FOIL, the Distributive Property or the Box Method/Area Model. Remember that i<sup>2</sup> = -1, i<sup>3</sup> = -i, i<sup>4</sup> = 1, i<sup>5</sup> = i, and so forth.
<CENTER>
<P>(5xi<sup>3</sup> - 4i)<sup>2</sup>
<BR>25x<sup>2</sup>i<sup>6</sup> - 40xi<sup>4</sup> + 16i<sup>2</sup>
<BR>-25x<sup>2</sup> - 40x - 16
</P>
</CENTER>
<P>
<BR><BR><BR>
<BR><em>30. Consider the parabola given by y = 1/4 x<sup>2</sup> + x + 8 vertex (-2,7) and focus (-2,8). Use this
information to explain how to determine the equation of the directrix.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>The focus is 1 unit above the vertex, so the directrix is the horizontal line that contains the point 1 unit below the vertex, (-2,6).
<P>The equation would be y = 6.
<P>Remember that you have to give an explanation, not just the equation. The equation alone is half-credit.
<P>
<BR><BR><BR>
<BR><em>31.Write (x √(x<sup>3</sup>) ) / (∛(x<sup>5</sup>)
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Remove the radicals by rewriting the expressions with fractional exponents. Then using the laws of exponents subtract the exponent of the numerator from the exponent of the denominator.
<CENTER><P>(x √(x<sup>3</sup>) ) / (∛(x<sup>5</sup>)</P>
<P>x (x <sup>(3/2)</sup>) / x <sup>(5/3)</sup></P>
<P>x <sup>(5/2)</sup> / x <sup>(5/3)</sup></P>
<P>x <sup>(5/2 - 5/3)</sup> </P>
<P>x <sup>(15/6 - 10/6)</sup> </P>
<P>x <sup>(5/6)</sup> </P>
</CENTER>
<P>
<BR><BR><BR>
<BR><em>32. A fruit fly population can be modeled by the equation P = 10(1.27)<sup>t</sup>, where P represents the number of fruit flies after t days. What is the average rate of change of the population, rounded to the nearest hundredth, over the interval [0,10.5]? Include appropriate units in your answer.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Find P(0) and P(10.5), subtract the latter from the initial value and then divide by 10.5.
<P>( 10(1.27)<sup>10.5</sup> - 10(1.27)<sup>0</sup>) / (10.5 - 0) = 10.7628...</P>
<P>10.76 fruit flies per day.</P>
<P>
<BR><BR><BR>
End of Part II
<P>How did you do?</P>
<BR><BR><BR>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-56543583540934874662024-02-03T17:23:00.001-05:002024-02-04T13:34:30.958-05:00June 2023 Algebra 2 Regents Part IV<font size=4 face="Times New Roman">
<BR><P>This exam was adminstered in June 2023. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>Algebra 2 June 2023</h3></u></em>
<P>Part IV: A correct answer will receive 6 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>37. The volume of air in an average lung during breathing can be modeled by the graph below.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbpBLnGKtlzuFVdezec44796zowiP8i70pwg4jw9Omkev13SB9yTG23kVGOm8FyQ5t-DtfwFTKNTebBEYbL8IKUNOIYJ5_50yqWnK7tLomZTcG9aNnkHfK5zpF2Jzj6HM5zCVJoCAiR794YyCs0QeeeCSqaD9CJiPL4To0Oa2pCBX1BpsibTiHBA/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="479" data-original-width="423" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbpBLnGKtlzuFVdezec44796zowiP8i70pwg4jw9Omkev13SB9yTG23kVGOm8FyQ5t-DtfwFTKNTebBEYbL8IKUNOIYJ5_50yqWnK7tLomZTcG9aNnkHfK5zpF2Jzj6HM5zCVJoCAiR794YyCs0QeeeCSqaD9CJiPL4To0Oa2pCBX1BpsibTiHBA/s1600/temp00.png"/></a></div>
<BR>Using the graph, write an equation for N(t), in the form N(t) = A sin (Bt) + C.
<P>That same lung, when engaged in exercise, has a volume that can be modeled by
E(t) = 2000 sin(πt) + 3200, where E(t) is volume in mL and t is time in seconds.
<p>Graph at least one cycle of E(t) on the same grid as N(t).
<p>How many times during the 5-second interval will N(t) = E(t)?
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR><I>A</I> is the height of the graph from the midline, which is 400. <I>B</I> is the period which is 2π/5 because it takes 5 seconds before 2 pi is graphed. C is the midline, which is 2400. Since the graph starts at the midline, it is a sine function, not a cosine function.
<P>The equation will be T = (400 - 75)e<sup>-.0735t</sup> + 75.</P>
<P>For the second part, look at the graph below. </P>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrOWUC-MeGP90Dzk7RDf8jnsj9ns54_MbAgBZDoqZLJWdVK-3HfaZJ2yKy-J5OzghPLoSRXH2REkNHSbZT1XDFJezdzmUX_GOeL8a90IdiDQI5maZ9DbAlwkeuEW_MBC4BE_5DE0lIHJpNGd7QGUoZ3-7KT57vJ_lhm3OtPBpIUKdRnQUiabdU-A/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="512" data-original-width="458" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrOWUC-MeGP90Dzk7RDf8jnsj9ns54_MbAgBZDoqZLJWdVK-3HfaZJ2yKy-J5OzghPLoSRXH2REkNHSbZT1XDFJezdzmUX_GOeL8a90IdiDQI5maZ9DbAlwkeuEW_MBC4BE_5DE0lIHJpNGd7QGUoZ3-7KT57vJ_lhm3OtPBpIUKdRnQUiabdU-A/s1600/temp00.png"/></a></div>
<P>In a 5-second interval, N(t) = E(t) four times.</P>
<P>
<BR><BR><BR>
End of Exam
<P>How did you do?</P>
<BR><BR><BR>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-23437274012117863732024-02-03T11:36:00.005-05:002024-02-04T13:34:01.976-05:00June 2023 Algebra 2 Regents, Part III<font size=4 face="Times New Roman">
<BR><P>This exam was adminstered in June 2023. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>Algebra 2 June 2023</h3></u></em>
<P>Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>33. Patricia creates a cubic polynomial function, p(x), with a leading coefficient of 1. The zeros of the
function are 2, 3, and -6. Write an equation for p(x).
<P>Sketch y = p(x) on the set of axes below.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>You are give the zeroes of the function and you need to have a 1 leading coefficient. You can write the function in factored form. You do not need to multiply it.
<P>p(x) = (x - 2)(x - 3)(x + 6) </P>
<P>If you aren't sure what this looks like, you can put it in your graphing calculator and look at it and the table of values.</P>
<P>When a cubic function has a positive leading coefficient, it starts at negative infinity and climbs to positive infinity. The end behavior is that as x goes to negative infinity, y goes to negative infinity, and when x goes to positive infinity, y goes to positive infinity. </P>
<P>You also have to have the graph cross the x-axis three times, at x = 2, x = 3 and x = -6.</P>
<P>Your graph should look like this:</P>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnjT3-NwKS-6VonLfARSTgLnMQuGzw9xYzp3ZsYIOvVnpCIJiVIJuXVKsu6zAKJJKlmOEyQC7cTElhr6eORMVXq0fP1wAIp3qfsFnmBEBPKEAKXrGkxY29TuEN-D8skrxjKqsCI8gMT65-mIMcClHYCkD1l3wKk8aPXEhkmhkOtDHm31VV_uDbhQ/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="317" data-original-width="406" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnjT3-NwKS-6VonLfARSTgLnMQuGzw9xYzp3ZsYIOvVnpCIJiVIJuXVKsu6zAKJJKlmOEyQC7cTElhr6eORMVXq0fP1wAIp3qfsFnmBEBPKEAKXrGkxY29TuEN-D8skrxjKqsCI8gMT65-mIMcClHYCkD1l3wKk8aPXEhkmhkOtDHm31VV_uDbhQ/s1600/temp00.png"/></a></div>
<P>
<BR><BR><BR>
<BR><em>34. A public radio station held a fund-raiser. The table below summarizes the donor category and
method of donation.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDRJrgByFl68QBfQZtoLkenR9axNdaEVFai8FsDfWELsMnzMRSPNWT7ujJDR1Xr6nZudNqOgH6v5m5_3vktVV085FAfckiwwfqF0gqqbKu3o05dLPaQyC9rwchYeD1iZvTHkwZU44ocsNrsBXGDmGTdA2qi1deuy4_ULXv3AQRPNYwdbzo-yuhfg/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="161" data-original-width="481" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDRJrgByFl68QBfQZtoLkenR9axNdaEVFai8FsDfWELsMnzMRSPNWT7ujJDR1Xr6nZudNqOgH6v5m5_3vktVV085FAfckiwwfqF0gqqbKu3o05dLPaQyC9rwchYeD1iZvTHkwZU44ocsNrsBXGDmGTdA2qi1deuy4_ULXv3AQRPNYwdbzo-yuhfg/s1600/temp00.png"/></a></div>
<P>To the nearest thousandth, find the probability that a randomly selected donor was categorized as
a supporter, given that the donation was made online.</P>
<P>Do these data indicate that being a supporter is independent of donating online? Justify your
answer. </P>
</em>
<P>
<P><B>Answer: </B> </P>
<br>There were 3216 donations made onlie. Of those, 1200 were from supporters. Probability is 1200/3216 = 0.3731..., or 0.373.
<P>There were a total of 1600 supporters out of 4286 donations made, and 1600/4286 = 0.373. So, yes, it is indepenpent because the probability of being a supporter is equal to the probability of being a supporter who donated online. </P>
<BR><BR><BR>
<BR><em>35. Algebraically solve the system:
<BR><CENTER>
(x - 2)<sup>2</sup> + (y - 3)<sup>2</sup> = 20<BR>y = -2x + 7
</CENTER><BR>
<BR>
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>The first equation is a circle and the second circle is a line. There may be 0, 1, or 2 possible solutions.
<P>Substitute -2x + 7 for y in the first equation and solve the quadratic equation.
<P><CENTER>
(x - 2)<sup>2</sup> + (-2x + 7 - 3)<sup>2</sup> = 20 <BR>
(x - 2)<sup>2</sup> + (-2x + 4)<sup>2</sup> = 20 <BR>
x<sup>2</sup> - 4x + 4 + 4x<sup>2</sup> - 16x + 16 = 20 <BR>
5x<sup>2</sup> - 20x + 20 = 20 <BR>
5x<sup>2</sup> - 20x = 0 <BR>
5x(x - 4) = 0<BR>
x = 0 or x = 4
</CENTER></P>
<P>Find the matching y-values.</P>
<P>y = -2(0) + 7 = 7, (0,7)
<BR>y = -2(4) + 7 = -1, (4,-1)</P>
<BR><BR><BR>
<BR><em>36. On a certain tropical island, there are currently 500 palm trees and 200 flamingos. Suppose the
palm tree population is decreasing at an annual rate of 3% per year and the flamingo population
is growing at a continuous rate of 2% per year.
<P>Write two functions, P(x) and F(x), that represent the number of palm trees and flamingos on this
island, respectively, x years from now.
<P>State the solution to the equation P(x) = F(x), rounded to the nearest year. Interpret the meaning
of this value within the given context.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Both functions are exponential. P can be written using the rate of decrease, compounded annually. Since F is continuous, e should be used.
<CENTER><P>P(x) = 500(1 - .03)<sup>x</sup> <BR> F(x) = 200e<sup>.02x</sup></P></CENTER>
<P>For the second part, you can graph the two functions and find the intersection point. Since it needs to be to the nearest year, you can look at the table of values.</P>
<P>The two will intersect around (18.159, 287.580)</P>
<P>There will be the same number of flamingoes and palm trees in 18 years.</P>
<P>
<BR><BR><BR>
End of Part III
<P>How did you do?</P>
<BR><BR><BR>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-671867071197330542024-02-01T13:18:00.003-05:002024-02-04T13:35:05.099-05:00June 2023 Algebra 2, Part II<font size=4 face="Times New Roman">
<BR>
<P>This exam was adminstered in June 2023. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>Algebra 2 June 2023</h3></u></em>
<P>Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>25. The business office of a local college wishes to determine the methods of payment that will be
used by students when buying books at the beginning of a semester. Explain how the office can
gather an appropriate sample that minimizes bias.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>The college should take a random sample survey from the list of students.
<P>Your answer should imply sort of randomization and include an appropriate sample.
<P>
<BR><BR><BR>
<BR><em>26. Determine the solution of √(3x + 7) = x - 1 algebraically.
<BR>
</em>
<P>
<P><B>Answer: </B> </P>
<br>Square both sides of the equation to get rid of the radical and then solve the equation that results from the exponents being equal.
<CENTER><P>√(3x + 7) = x - 1
<BR>3x + 7 = x<sup>2</sup> - 2x + 1
<BR>0 = x<sup>2</sup> - 5x - 6
<BR>0 = (x - 6)(x + 1)
<BR>x = 6 or x = -1<P></CENTER>
<P>Check both answers:</P>
<P>√(3(6) + 7) = √(25) = 5; 6 - 1 = 5, Check.
<BR>√(3(-1) + 7) = √(4) = 2; -1 - 1 = -2. discard as extraneous.</P>
<P>x = 6 is the only answer.
<BR><BR><BR>
<BR><em>27. The population of bacteria, P(t), in hundreds, after t hours can be modeled by the function
P(t) = 37e<sup>0.0532t</sup>
. Determine whether the population is increasing or decreasing over time. Explain
your reasoning.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>You can graph this with your calculator and state that the graph is rising so the population is increasing. Or you can evaluate e<sup>0.0532t</sup>.
<P>Since e<sup>0.0532t</sup> = 1.0546..., which is greater than 1, the population is increasing.</P>
<BR><BR><BR>
<BR><em>28. The polynomial function g(x) = x<sup>3</sup> + ax<sup>2</sup> - 5x + 6 has a factor of (x - 3). Determine the value of a.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Set the function equal to 0, substitute x = 3, and solve for a.
<CENTER>
<P>g(3) = (3)<sup>3</sup> + a(3)<sup>2</sup> - 5(3) + 6 = 0 <BR>
27 + a(9) - 15 + 6 = 0 <BR>
9a = - 18 <BR>
a = -2</P>
</CENTER>
<P>
<BR><BR><BR>
<BR><em>29. Write a recursive formula for the sequence 189, 63, 21, 7, … .
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>A recursive function needs and a<sub>1</sub> value followed by a<sub>n</sub> defined in terms of a<sub>n-1</sub>.
<P>a<sub>1</sub> = 189<BR>a<sub>n</sub> = (1/3) a<sub>n-1</sub>
</P>
<P>
<BR><BR><BR>
<BR><em>30. Solve algebraically for x to the nearest thousandth:
<CENTER>2e<sup>0.49x</sup> = 15
</CENTER>
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Solve using inverse operations, including natural logs.
<CENTER>
<P>2e<sup>0.49x</sup> = 15
<br>e<sup>0.49x</sup> = 15/2
<br>ln(e<sup>0.49x</sup>) = ln(15/2)
<br>0.49x = 2.01490...
<br>x = 4.112
</CENTER>
<P>
<BR><BR><BR>
<BR><em>31. For all values of x for which the expression is defined, write the expression below in simplest form.
<BR><CENTER>(2x<sup>3</sup> + x<sup>2</sup> - 18x - 9) / (3x - x<sup>2</sup>)
</CENTER>
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Factor the numerator and the denominator and cancel out the common factors.
<CENTER><P>(2x<sup>3</sup> + x<sup>2</sup> - 8x - 9) / (3x - x<sup>2</sup>)</P>
<P>( x<sup>2</sup>(2x + 1) - 9(2x + 1) ) / ( (x)(3 - x) ) </P>
<P>( (x<sup>2</sup>- 9)(2x + 1) ) / ( (x)(3 - x) ) </P>
<P>( (x + 3)(x - 3)(2x + 1) ) / ( (x)(-1)(x - 3) ) </P>
<P>( (x + 3)(2x + 1) ) / ( (-x) ) </P>
</CENTER>
<P>
<BR><BR><BR>
<BR><em>32. An app design company believes that the proportion of high school students who have purchased
apps on their smartphones in the past 3 months is 0.85. A simulation of 500 samples of 150 students
was run based on this proportion and the results are shown below.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhdxB46BA0P3YN0EpK_a-hM-shoroARtNTs-k5bdVDuqPCXudsoBW9lUcLFhyphenhyphengN12mEoAPu-Sq30K2u6SXrt2sTaDUCjGqcS6C_YiVNRtIdqjlmpkxTCEht8x521PVFslH9xioZjmvAxOgGGnFUHmos7pC35omCcy3F9FFmlSoPH7QKxmeUZeGNw/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="346" data-original-width="516" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhdxB46BA0P3YN0EpK_a-hM-shoroARtNTs-k5bdVDuqPCXudsoBW9lUcLFhyphenhyphengN12mEoAPu-Sq30K2u6SXrt2sTaDUCjGqcS6C_YiVNRtIdqjlmpkxTCEht8x521PVFslH9xioZjmvAxOgGGnFUHmos7pC35omCcy3F9FFmlSoPH7QKxmeUZeGNw/s1600/temp00.png"/></a></div>
<P>Suppose a sample of 150 students from your high school showed that 88% of students had
purchased apps on their smartphones in the past 3 months. Based on the simulation, would the
results from your high school give the app design company reason to believe their assumption is
incorrect? Explain.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Find the confidence interval by multiplying the standard deviation (SD) by 2. Subtract that from the mean to get the lower bound of the interval and add it to the mean to get the upper bound. See if 88% falls within that interval.
<P>(2)(0.029) = 0.058</P>
<P>Lower bound = 0.852 - 0.058 = 0.794. Upper bound = 0.852 + 0.058 = 0.910.</P>
<P>Since 88% falls inside the confidence interval, the results would not give the company reason to believe that their assumption is incorrect.
<P>
<BR><BR><BR>
End of Part II
<P>How did you do?</P>
<BR><BR><BR>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-73101648894183428812024-01-31T10:25:00.002-05:002024-02-04T13:35:19.711-05:00January 2023 Algebra 2 Regents Part IV<font size=4 face="Times New Roman">
<BR><P>This exam was adminstered in January 2023. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>Algebra 2 January 2023</h3></u></em>
<P>Part IV: A correct answer will receive 6 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>37. A Objects cool at different rates based on the formula below.
<CENTER>
T = (T<sub>0</sub> - T<sub>R</sub>)e<sup>2rt</sup> + T<sub>R</sub>
T<sub>0</sub>: initial temperature
T<sub>R</sub>: room temperature
r: rate of cooling of the object
t: time in minutes that the object cools to a temperature, T</CENTER>
<BR>
Mark makes T-shirts using a hot press to transfer designs to the shirts. He removes a shirt from
a press that heats the shirt to 400°F. The rate of cooling for the shirt is 0.0735 and the room
temperature is 75°F. Using this information, write an equation for the temperature of the shirt,
T, after t minutes.
<BR>Use the equation to find the temperature of the shirt, to the nearest degree, after five minutes.
<BR>At the same time, Mark’s friend Jeanine removes a hoodie from a press that heats the hoodie to
450°F. After eight minutes, the hoodie measured 270°F. The room temperature is still 75°F.
Determine the rate of cooling of the hoodie, to the nearest ten thousandth.
<BR>The T-shirt and hoodie were removed at the same time. Determine when the temperature will be
the same, to the nearest minute.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Write the equation substituting all values that we know. You should only have <i>t</i> remaining (and the letter <i>e</i> -- don't replace that with a number).
<P>The equation will be T = (400 - 75)e<sup>-.0735t</sup> + 75.</P>
<P>For the second part, substitute t = 5 and evaluate in your calculator:
<BR>T = (400 - 75)e<sup>-.0735(5)</sup> + 75 = 300.05058... = 300 degrees </P>
<P>In the next part, you are given T and t but you need to find r:
<BR>270 = (450 - 75)e<sup>-r(8)</sup> + 75
<BR>195 = (375)e<sup>-8r</sup>
<BR>195/375 = e<sup>-8r</sup></P>
<P>log<sub>e</sub> 195/375 = -8r<br>r = (log<sub>e</sub> 195/375)/(-8) <BR> r = 0.08174... = 0.0817.</P>
<P>For the last piece, we need to find t when the two expressions will be equal:
<BR>(375)e<sup>-0.0817t</sup> + 75 = (325)e<sup>-0.0735t</sup> + 75
<BR>(375)e<sup>-0.0817t</sup> = (325)e<sup>-0.0735t</sup>
<BR>e<sup>-0.0817t</sup> = (325/375)e<sup>-0.0735t</sup>
<BR>e<sup>-0.0817t</sup> / e<sup>-0.0735t</sup> = 13/15
<BR>e<sup>-0.0082t</sup> = 13/15
<BR>ln 13/15 = -0.0082t
<BR>t = (ln 13/15) / (-0.0082) = 17.451...</P>
<P>About 17 minutes.</P>
<P>You could also have plugged each equation into your graphing calculator and compared the tables of values of the two equations. The intersection would happen at approximately 17 minutes. This will receive full credit if you explain where you got the answer from and likely allows fewer opportunites for mistakes.</P>
<P>As it is, I worked it out just to be sure that I could work it out. And in the end, I had an incorrect answer from putting the last step into the calculator incorrectly. Thankfully, I double-checked my work and discovered an error. Then I had to figure which of the two was incorrect!</P>
<P>
<BR><BR><BR>
End of Exam
<P>How did you do?</P>
<BR><BR><BR>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-71091740461625074102024-01-30T18:44:00.002-05:002024-01-30T18:44:42.033-05:00School Life #39: Helium and Iron<font size="2">(Click on the comic if you can't see the full image.)</font><br />
<center>
<aa href="http://xwhy.comicgenesis.com/d/20240130.html">
<iii src="http://mrburkemath.net/xwhy/images/1776 .png" title="When you give all you have to give, that's giving 100%. The math checks. "></iii>
</aa>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp8dOAWfE_r3moQ5qawZLsBC96le3rVui3QvEa-G3KjW8jGghSJFybwUW0HZFDNA7ZFNDRlvumOgIVVWM9OE0dInA8DZDzLCZCVr-eH4vXie0XkQR1J9BnYojfw44YOQFwP6KwI-be4OGKKfq1C-ZvWMKh3pa2cN3dYYsCgHu5YFCsjZA_y_7jTw/s1600/schoollife40-HE-FE.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="350" data-original-width="1000" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp8dOAWfE_r3moQ5qawZLsBC96le3rVui3QvEa-G3KjW8jGghSJFybwUW0HZFDNA7ZFNDRlvumOgIVVWM9OE0dInA8DZDzLCZCVr-eH4vXie0XkQR1J9BnYojfw44YOQFwP6KwI-be4OGKKfq1C-ZvWMKh3pa2cN3dYYsCgHu5YFCsjZA_y_7jTw/s1600/schoollife40-HE-FE.png"/></a></div>
<font color="FF4444">(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).</font>
</center>
<font size=4>
<P>Learning was assessed, just not the learning that was expected.</P>
<P>This was just one of those things that pops up in Geometry class when things are written on the board, and they appear to be something else when looked at out of context. And someone who is less aware (or more asleep) might not have the proper context. (Reference "What's a Parabola?")</P>
<P>A new semester is under way. I hope it's fruitful. In the meantime, I'm catching up on old Regents exams so I can avoid rewriting something that needs to be rewritten even though I don't want to rewrite it. (Edit, sure, but rewrite?) </P>
</font>
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
<br /><br /><i>Come back often for more funny math and geeky comics.</i>
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<br />(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-38535451035594394972024-01-30T08:34:00.002-05:002024-02-04T13:35:37.461-05:00January 2023 Algebra 2 Regents, Part III<font size=4 face="Times New Roman">
<BR><P>This exam was adminstered in January 2023. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>Algebra 2 January 2023</h3></u></em>
<P>Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>33. Solve the equation √(49 - 10x) + 5 = 2x algebraically.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Subtract 5 from both sides to isolate the radical. Square both sides. Then solve the quadratic equation. Finally, discard extraneous solutions that were added when you squared the equation.
<P>You do this by checking the answers that you got. </P>
<CENTER>
<P>√(49 - 10x) + 5 = 2x</P>
<P>√(49 - 10x) = 2x - 5</P>
<P>49 - 10x = 4x<sup>2</sup> - 20x + 25</P>
<P>0 = 4x<sup>2</sup> - 10x - 24 </P>
<P>0 = 2x<sup>2</sup> - 5x - 12 </P>
<P>0 = (2x + 3)(x - 4)</P>
<P>2x + 3 = 0 or x - 4 = 0 </P>
<P>x = -3/2 or x = 4</P>
</CENTER>
<P>Check -3/2: √(49 - 10(-3/2)) + 5 ?= 2(-3/2)
<BR>√(49 - 10(-3/2) ?= -3 - 5 = -8
<BR>The square root cannot be negative, so <b>reject</b> this answer.</P>
<P>Check 42: √(49 - 10(4)) + 5 ?= 2(4)
<BR>√(9) + 5 ?= 8
<BR>3 + 5 = 8 (check!)
<BR>x = 4 is the only solution.</P>
<P>
<BR><BR><BR>
<BR><em>34. Joette is playing a carnival game. To win a prize, one has to correctly guess which of five equally sized regions a spinner will land on, as shown in the diagram below.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp1E_LBgZVy52KFub6m_g3hZUG-XP_2hEXNLj_iy2pcSjVjM6Wli3Ri8MAV5CgTudBy2Quff6JMWWOz2a93suAQRsot62OUTg1nUqzgY9FErJIdYuRxqcqQH8BNbZz3ymqnD3OD9jfElLGyctS9ljZeUYrBQ-KUfJHrBfNt0XiDsduVQS4Rawu4Q/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="154" data-original-width="238" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp1E_LBgZVy52KFub6m_g3hZUG-XP_2hEXNLj_iy2pcSjVjM6Wli3Ri8MAV5CgTudBy2Quff6JMWWOz2a93suAQRsot62OUTg1nUqzgY9FErJIdYuRxqcqQH8BNbZz3ymqnD3OD9jfElLGyctS9ljZeUYrBQ-KUfJHrBfNt0XiDsduVQS4Rawu4Q/s1600/temp00.png"/></a></div>
<P>She complains that the game is unfair because her favorite number, 2, has only been spun once in ten times she played the game.
<P>State the proportion of 2’s that were spun.
<P>State the theoretical probability of spinning a 2.
<BR>
<P> The simulation output below shows the results of simulating ten spins of a fair spinner, repeated 100 times.</P>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZBCMnGaYQDxcVCLrEjPqzDjov9ju5gnH-femkFyz2zhl3_-qZEE9ZXGjlXuyvJ7upKgLiooWCh4KUOuyjWrg9MqX2bQlJyaJZyqGodCXteQOFlj8ukFlJJmIdEkUBD3ek6hjJlf0-n3BDYNVvf2w9WNV9iVgzYdGdstsgWEmGRPc37m3VvSYy3w/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="338" data-original-width="420" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZBCMnGaYQDxcVCLrEjPqzDjov9ju5gnH-femkFyz2zhl3_-qZEE9ZXGjlXuyvJ7upKgLiooWCh4KUOuyjWrg9MqX2bQlJyaJZyqGodCXteQOFlj8ukFlJJmIdEkUBD3ek6hjJlf0-n3BDYNVvf2w9WNV9iVgzYdGdstsgWEmGRPc37m3VvSYy3w/s1600/temp00.png"/></a></div>
<P>Does the output indicate that the carnival game was unfair? Explain your answer. </P>
</em>
<P>
<P><B>Answer: </B> </P>
<br>Empirical probability is <b>1/10</b> because there was 1 positive result in 10 tries.
<P>Theoretical probability is <b>1/5</b> because the number 2 covers one-fifth of the area of the wheel. There are five possible outcome and each should be as likely to occur as the others.</P>
<P>The chart shows that in 21 out of 100 simulations a result of 1/10, or 0.10 occured. This is not an unusual outcome.</P>
<P>If you think about it, with 5 possible outcomes, in 10 spins, a result of 2 should happen two times. If a result of 2 only happened one time, or even 3 times, in 10 spins that wouldn't be an unlikely occurrence because variations can be expected.</P>
<BR><BR><BR>
<BR><em>35. Graph c(x) = -9(3)<sup>x - 4</sup> + 2 on the axes below.
<BR>Describe the end behavior of c(x) as x approaches positive infinity.
<BR>Describe the end behavior of c(x) as x approaches negative infinity.
<BR>
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>You can use your graphing calculator to get the table of values for where to plot the points. Your graph should look like this one below.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkYFFh6MTNi8beV9jx3FiigRHh-kjJNgb7huR6QgLWdFurdn5sUKLyTUAp3pgvbUS1CJwjKhehkmfPPK9VCPkcVKPn9k2jzGAqybufu4eTwfsNpgN05rOJq3Aa6ahmFX-q6l6n90hfq_oX3rgoguvcNFIEY2I6QBKfuBCh0fPLt8e32l9ehkRpFQ/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="563" data-original-width="537" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkYFFh6MTNi8beV9jx3FiigRHh-kjJNgb7huR6QgLWdFurdn5sUKLyTUAp3pgvbUS1CJwjKhehkmfPPK9VCPkcVKPn9k2jzGAqybufu4eTwfsNpgN05rOJq3Aa6ahmFX-q6l6n90hfq_oX3rgoguvcNFIEY2I6QBKfuBCh0fPLt8e32l9ehkRpFQ/s1600/temp00.png"/></a></div>
<P>The end behavior of c(x): as x approaches positive infinity, c(x) approaches negative infinity. As x approaches negative infinity, c(x) approaches 2.</P>
<BR><BR><BR>
<BR><em>36. The monthly high temperature (°F) in Buffalo, New York can be modeled by
B(m) = 24.9sin(0.5m - 2.05) + 55.25, where m is the number of the month and January = 1.
<BR>Find the average rate of change in the monthly high temperature between June and October, to
the nearest hundredth.
<BR>Explain what this value represents in the given context.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>June is the 6th month and October is the 10th month. Find B(6) and B(10). Subtract B(10) - B(6) and divide the difference by (10 - 6), which is 4 to find the average monthly change.
<P>B(6) = 24.9*sin(0.5(6) - 2.05) + 55.25 = 75.5040
<br>B(10) = 24.9*sin(0.5(10) - 2.05) + 55.25 = 59.9915</P>
<P>(B(10) - B(6)) / 4 = (59.9915 - 75.5040) / 4 = -3.878 = -3.88</P>
<P>This means that for each month, the temperature is decreasing by 3.88 degrees.</P>
<P>
<BR><BR><BR>
End of Part III
<P>How did you do?</P>
<BR><BR><BR>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-20553411031623800632024-01-29T02:00:00.006-05:002024-02-04T13:35:53.661-05:00January 2023 Algebra 2, Part II<font size=4 face="Times New Roman">
<BR>
<P>This exam was adminstered in January 2023. </P>
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<P><em><u><h3>Algebra 2 January 2023</h3></u></em>
<P>Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.
<BR><BR>
<BR><em>25. Algebraically determine the zeros of the function below.
<CENTER>r(x) = 3x<sup>3</sup> + 12x<sup>2</sup> - 3x - 12</CENTER>
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Set the expression equal to 0. Then factor by grouping.
<CENTER><P>
3x<sup>3</sup> + 12x<sup>2</sup> - 3x - 12 = 0</P>
<P>3x<sup>3</sup> - 3x + 12x<sup>2</sup> - 12 = 0</P>
<P>3x(x<sup>2</sup> - 1) + 12(x<sup>2</sup> - 1) = 0</P>
<P>(3x + 12)(x<sup>2</sup> - 1) = 0</P>
<P>(3x + 12)(x - 1)(x + 1) = 0</P>
<P>3x + 12 = 0 or x - 1 = 0 or x + 1 = 0 </P>
<P>x = -4 or x = 1 or x = -1</P>
</CENTER>
<P>
<BR><BR><BR>
<BR><em>26. Given a > 0, solve the equation a<sup>x + 1</sup> = ∛(a<sup>2</sup>) for x algebraically
<BR>
</em>
<P>
<P><B>Answer: </B> </P>
<br>Cube both sides of the equation to get rid of the radical and then solve the equation that results from the exponents being equal.
<CENTER><P>a<sup>x + 1</sup> = ∛(a<sup>2</sup>)
<BR>a<sup>3x + 3</sup> = a<sup>2</sup>
<BR>3x + 3 = 2
<BR>3x = -1
<BR>x = -1/3</P></CENTER>
<BR><BR><BR>
<BR><em>27. Given P(A) = 1/3 and P(B) = 5/12, where A and B are independent events, determine P(A ∩ B).
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>The probably of two independent events happening in the probability of one of them happening times the probability of the other.
<P>P(A ∩ B) = (1/3)(5/12) = 5/36.</P>
<BR><BR><BR>
<BR><em>28. The scores on a collegiate mathematics readiness assessment are approximately normally distributed with a mean of 680 and a standard deviation of 120.
<P>Determine the percentage of scores between 690 and 900, to the nearest percent.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Use your graphing calculator. Find the function normalcdf.
<P>Enter the following command: normalcdf(690,900,680,120) for the range minimum and maximum, followed by the median and the standard deviation. The answer will be 43%.</P>
<P>If you estimate it using a standard deviation chart, you won't get an exact answer.
<P>
<BR><BR><BR>
<BR><em>29. Consider the data in the table below.
<P><TABLE border=1 cellpadding=4><tr>
<td>x</td><td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td>
</tr>
<tr>
<td>y</td><td>3.9</td> <td>6</td> <td>11</td> <td>18.1</td> <td>28</td> <td>40.3</td>
</tr>
</TABLE>
<P>State an exponential regression equation to model these data, rounding all values to the nearest
thousandth.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Put the data into List 1 and List 2 on your graphing calculator. Run a Exponential Regression (ExpReg).
<P>You should get the following output: a = 2.4585... and b = 1.6159...
</P>
<P>The equation you want is y = (2.459)(1.616)<sup>x</sup></P>
<P>
<BR><BR><BR>
<BR><em>30. Write the expression A(x) • B(x) - 3C(x) as a polynomial in standard form.
<CENTER>A(x) = x<sup>3</sup> + 2x - 1
<BR>B(x) = x<sup>2</sup> + 7
<BR>C(x) = x<sup>4</sup> - 5x
</CENTER>
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Multiply the first two expression A(x) and B(x). Subtract the product of 3 times C(x).
<CENTER>
<P>(x<sup>3</sup> + 2x - 1)(x<sup>2</sup> + 7) - 3(x<sup>4</sup> - 5x)
<br>x<sup>5</sup> + 7x<sup>3</sup> + 2x<sup>3</sup> + 14x - x<sup>2</sup> - 7 - 3x<sup>4</sup> + 15x
<br>x<sup>5</sup> + 9x<sup>3</sup> - x<sup>2</sup> + 14x - 7 - 3x<sup>4</sup> + 15x
<br>x<sup>5</sup> - 3x<sup>4</sup> + 9x<sup>3</sup> - x<sup>2</sup> + 29x - 7
</CENTER>
<P>
<BR><BR><BR>
<BR><em>31. Over the set of integers, completely factor x<sup>4</sup> - 5x<sup>2</sup> + 4
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>The first step is to factor it the way you would factor y<sup>2</sup> - 5y + 4. Then factor the quadratic expression that result.
<CENTER><P>x<sup>4</sup> - 5x<sup>2</sup> + 4
<P>(x<sup>2</sup> - 4)(x<sup>2</sup> - 1)</P>
<P>(x - 2)(x + 2)(x - 1)(x + 1)</P></CENTER>
<P>
<BR><BR><BR>
<BR><em>32. Natalia’s teacher has given her the following information about angle θ.
<CENTER>
• π < θ < 2π
• cos θ = &sqrt;(3)/4</CENTER>
<P>Explain how Natalia can determine if the value of tan θ is positive or negative.
</em>
<BR>
<P>
<P><B>Answer: </B> </P>
<BR>Cosine is positive in Quadrants I and IV, and π < θ < 2π indicates Quadrants III and IV, so θ must put the angle into quadrant IV.
<P>Tangent is negative in Quadrant IV.
<P>
<BR><BR><BR>
End of Part II
<P>How did you do?</P>
<BR><BR><BR>
<BR><BR><BR>
<BR>More to come. Comments and questions welcome.
<P>More <a href="https://mrburkemath.blogspot.com/search/label/Regents">Regents problems</a>.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-24833708932551313642024-01-28T13:02:00.002-05:002024-02-04T13:36:46.577-05:00August 2023 Geometry Regents Part IV
<BR><P>This exam was adminstered in August 2023. </P>
<!-- <BR>The answers to Part I can be found <a href="https://mrburkemath.blogspot.com/2019/04/january-2019-common-core-geometry.html">here</a>
<BR>The answers to Parts III and IV can be found <a href="https://mrburkemath.blogspot.com/2019/03/january-2019-common-core-geometry.html">here</a> -->
<font size=4 face=helvetica>
<P><h3>August 2023 Geometry, Part IV</h3>
<P>A correct answer is worth 6 credits. Partial credit can be given for correct statements in the proof.
<BR><BR><BR><P><strong>35.</strong> <em>In the diagram below of quadrilateral FACT, BR intersects diagonal AT at E,
AF || CT, and AF ≅ CT.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNDRMCkPyCEUzdwMB320Q7r2cYOigVBqv7wTsJ5YizG4Ja7vWIkADjSrC_Txsvpi3NnYrmIjqdTb4Blt-N3ZAg2eQ58V-W7ljr1dO4yn3dVepfRlOI79hEQHcm6h37p4071ME6qIQ5XX85DL2OOryP3uem3uV2znn6r6skLTMdTVw_EYAj4pPhiA/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="286" data-original-width="552" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNDRMCkPyCEUzdwMB320Q7r2cYOigVBqv7wTsJ5YizG4Ja7vWIkADjSrC_Txsvpi3NnYrmIjqdTb4Blt-N3ZAg2eQ58V-W7ljr1dO4yn3dVepfRlOI79hEQHcm6h37p4071ME6qIQ5XX85DL2OOryP3uem3uV2znn6r6skLTMdTVw_EYAj4pPhiA/s1600/temp00.png"/></a></div>
<P>Prove: (AB)(TE) = (AE)(TR)
</em>
<P><b>Answer: </b> </P>
<BR>This question is a little different, because most proofs of this type usuablly rely on find two traingles to be congruent via SSS, SAS, etc., and then showing two parts to be congruent using CPCTC. This one doesn't do that. This one asks you to prove a multiplication statement is true, but these products are the results of cross-multiplying the numerators and denominators of a proportion. You need to show that the two triangles are similar, not congruent, and then cross multiply the sides because the product of the means will equal the product of the extremes.
<P>Also, notice that you are only told that this is a "quadrilateral", not a parallelogram. This was the case in June, as well. I wonder if it's going to be a recurring theme going forward.</P>
<P>Your proof should look like this:</P>
<P>
<center>
<table width=90% border=5 cellspacing = 2 cellpadding = 2>
<tr>
<td width = 40%>Statement
</td>
<td>Reasons
</td>
</tr>
<tr>
<td>Quadrilateral FACT, BR intersects diagonal AT at E,
AF || CT, and AF ≅ CT.
</td>
<td>Given
</td>
</tr>
<tr>
<td>FACT is a parallelogram
</td>
<td>A quadrilateral with one pair of sides that are parallel and congruent is a parallelogram
</td>
</tr>
<tr>
<td>∠ BAE ≅ ∠ RTE
</td>
<td>Alternate Interior Angles
</td>
</tr>
<tr>
<td>∠ AEB ≅ ∠ TER
</td>
<td>Vertical Angles
</td>
</tr>
<tr>
<td>△AEB ~ △RET
</td>
<td>AA Postulate
</td>
</tr>
<tr>
<td>(AB) / (AE) = (TR) / (TE)
</td>
<td>Corresponding sides of similar triangles are proportional
</td>
</tr>
<tr>
<td>(AB)(TE) = (AE)(TR)
</td>
<td>The product of the means = the product of the extremes
</td>
</tr>
</table>
</center>
</P>
<P>Each statement is important. If you leave any out, you will lose one credit. However, if you have a couple of statements correct and the semblance of a proof, you should still earn two points for the question.</P>
<P>Also note that I wrote "The product of the means = the product of the extremes" instead of "cross multiplication", which is in essence the same thing, except that it is an instruction and not a justification or Reason (theorem, postualte, etc.). You could argue this point all you want, but there is an example in the Model Response Set where the only point lost is for the final step where "Cross multiply" is written, and the Regents literally calls out this step as an incorrect reason.</P>
<BR><BR><BR>
<P><B>End of Exam</B>
<P>How did you do?
<P>Questions, comments and corrections welcome.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-32165522332004166432024-01-27T12:33:00.003-05:002024-02-04T13:37:01.357-05:00August 2023 Geometry Regents Part III
<BR><P>This exam was adminstered in August 2023. </P>
<!-- <BR>The answers to Part I can be found <a href="https://mrburkemath.blogspot.com/2019/04/january-2019-common-core-geometry.html">here</a>
<BR>The answers to Parts III and IV can be found <a href="https://mrburkemath.blogspot.com/2019/03/january-2019-common-core-geometry.html">here</a> -->
<font size=4 face=helvetica>
<P><h3>August 2023 Geometry, Part III</h3>
<P>Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.
<BR><BR><BR><P><strong>32.</strong> <em>
Josh is making a square-based fire pit out of concrete for his backyard, as modeled by the right
prism below. He plans to make the outside walls of the fire pit 3.5 feet on each side with a height
of 1.5 feet. The concrete walls of the fire pit are going to be 9 inches thick.
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<BR> If a bag of concrete mix will fill 0.6 ft<sup>3</sup>, determine and state the minimum number of bags needed
to build the fire pit.
</em>
<P><b>Answer: </b> </P>
<BR>First you need to find the Volume, which is length * width * height (rember that it has a sqware base) minus the volume of the hole in the center, which has the same height but is 18 inches less wide and long. Also keep in mind that they are mixing feet and inches in this problem. (Guess what! Real life does that, too!)
<P>Nine inches is .75 feet, and 18 inches is 1.5 feet. Thus, 3.5 - 1.5 = 2.0 feet.
<P>Volume = (3.5)(3.5)(1.5) - (2)(2)(1.5) = 12.375 ft<sup>3</sup></P>
<P>If one bag of concrete mix fills 0.6 ft<sup>3</sup>, then the required number will be 12.375/0.6 = 20.625 or 21 when rounded up. You must round up or you will not have enough concrete mix to complete the job. </P>
<BR><BR><BR><P><strong>33.</strong> <em> A telephone pole 11 meters tall needs to be stabilized with a support beam, as modeled below.
<P>Two conditions for proper support are:
<P>• The beam reaches the telephone pole at 70% of the telephone pole’s height above the ground.
<P>• The beam forms a 65° angle with the ground.
<P>Determine and state, to the nearest tenth of a meter, the length of the support beam that meets these conditions for this telephone pole.
<P>Determine and state, to the nearest tenth of a meter, how far the support beam must be placed from the base of the pole to meet the conditions.
</em>
<P><b>Answer: </b></P>
<BR>First, find 70% of 11 meters (by multiplying) to find how far up the pole the beam will reach. Then use the sin 65° to find the length of the beam, and either tan 65° or the Pythagorean Theorem to find the distance on the ground.
<P>11 * .7 = 7.7 meters.</P>
<P>sin 65° = 7.7 / x <BR> x = 7.7 / sin 65° = 8.46 = 8.5 meters</P>
<P>tan 65° = 7.7 / y <BR> y = 7.7 / tan 65° = 3.59 = 3.6 meters</P>
<P>Look at these answers. The beam is 8.5 meters, which is reasonable for the hypotenuse when the longer leg is 7.7. And the base is 3.6 meters, which is less than half of 7.7, which is reasonable for the side opposite a 25 degree angle. If the numbers you got seem extremely large or extremely small, you may have done one of three things: used radians instead of degrees, entered the ratio incorrectly (or used the wrong one), or forgot a decimal point somewhere.</P>
<P>If you got a <i>negative</i> answer, it's probably radians.</P>
<BR><BR><BR><P><strong>34.</strong> <em> The coordinates of the vertices of quadrilateral ABCD are A(0,4), B(3,8), C(8,3), and D(5,-1).
<P>Prove that ABCD is a parallelogram, but not a rectangle.
<BR>[The use of the set of axes below is optional.]
<BR>
</em>
<P><b>Answer: </b></P>
<BR>Use the grid if you want to visualize the problem better. You can also use it to help with finding slopes.
<P>A parallelogram has two pairs of parallel sides, and those sides will therefore have the same slope. In a rectangle, the sides must be perpendicular, which means that the slopes are inverse reciprocals and that the products of the slopes of the two lines will be -1.</P>
<P>Find all four slopes. Show that the opposite sides have the same slope. Show that the consecutive sides are not perpendicular. Then write your concluding statement.</P>
<P>Slope of AB = (8-4) / (3-0) = 4/3. Slope of BC = (3-8) / (8-3) = -5/5 = -1. <BR>Slope of CD = (-1-3) / (5-8) = -4/-3 = 4/3. Slope of DA = (4-(-1)) / (0-5) = 5/-5 = -1.</P>
<P>AB || CD and BC || DA because they have the same slopes, therefore ABCD is a parallelogram.</P>
<P>Slope of AB * Slope of BC = (4/3)(-1) = -4/3. The slopes are not inverse reciprocals, so the lines are not perpendicular. Therefore, ABCD is NOT a rectangle.</P>
<i><P>Anther method:</P></i>
<P>Paralelograms have a lot of properties, as do rectangles. If you don't mind use the distance formula (Pythagorean Theorem, if you graphed ABCD), you can show that the opposites sides are congruent, which makes it a parallelogram, but the diagonals are NOT congruent, which means is could not be a rectangle.</P>
<P>This is a valid option. However, I'd prefer to use the slope formula over the distance formula is I could. You basically have to find the slope to use the distance formula.</P>
<P><B>End of Part III</b>
<P>How did you do?
<P>Questions, comments and corrections welcome.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0tag:blogger.com,1999:blog-28172905.post-8507195456788613962024-01-26T19:59:00.001-05:002024-02-04T13:37:15.149-05:00August 2023 Geometry Regents Part II
<BR><P>This exam was adminstered in August 2023. </P>
<!-- <BR>The answers to Part I can be found <a href="https://mrburkemath.blogspot.com/2019/04/january-2019-common-core-geometry.html">here</a>
<BR>The answers to Parts III and IV can be found <a href="https://mrburkemath.blogspot.com/2019/03/january-2019-common-core-geometry.html">here</a> -->
<font size=4 face=helvetica>
<P><h3>August 2023 Geometry, Part II</h3>
<P>Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.
<BR><BR><BR><P><strong>25.</strong> <em>
On the set of axes below, congruent quadrilaterals ROCK and R'O'C'K' are graphed.
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<P>Describe a sequence of transformations that would map quadrilateral ROCK onto
quadrilateral R'O'C'K'.
</em>
<P><b>Answer: </b> </P>
<BR>The figure has to move from Quadrant IV to Quadrant II. Inspection shows that the orientation of the quadrilateral has rotated. However, a rotation about the origin will not create the image of K at (-1,1), so a translation must follow.
<P>Rotate ROCK 180 degrees around the origin. Then translate the image 2 to the left and 1 up.</P>
<P>There are many possible answers involding rotations around different centers, or reflections across the x-axis and y-axis or even other lines.</P>
<BR><BR><BR><P><strong>26.</strong> <em> In triangle CEM, CE = 3x + 10, ME = 5x - 14, and CM = 2x - 6.
Determine and state the value of x that would make △CEM an isosceles triangle with
the vertex angle at E.
</em>
<P><b>Answer: </b></P>
<BR>If the vertex angle is E then sides CE and ME must be congruent and their lengths are equal.
<P>Therefore,
<CENTER>5x - 14 = 3x + 10 <BR>2x = 24 <BR> x = 12 </CENTER>
<BR><BR><BR><P><strong>27.</strong> <em>A flagpole casts a shadow on the ground 91 feet long, with a 53° angle of elevation from the end
of the shadow to the top of the flagpole.
<P>Determine and state, to the nearest tenth of a foot, the height of the flagpole.
</em>
<P><b>Answer: </b></P>
<BR>The height and the shadow mean that we need to use tangent. (We aren't concerned with the hypotenuse of the right triangle created.
<P>The height is opposite the 53° angle and the shadow on the ground is adjacent to it.</P>
<P><CENTER>tan 53 = x / 91 <BR> x = 91 tan 53 = 120.76... = 120.8 feet</CENTER></P>
<BR><BR><BR><P><strong>28.</strong> <em>A man is spray-painting the tops of 10 patio tables. Five tables have round tops, with diameters
of 4 feet, and five tables have rectangular tops, with dimensions of 4 feet by 6 feet. A can of spray
paint covers 25 square feet. How many cans of spray paint must be purchased to paint all of the
tabletops?
</em>
<P><b>Answer: </b></P>
<BR>Find the area of one circle (πr<sup>2</sup>) and multiply by 5. Find the area of one rectangle (A=LW) and multiply by 5.
<P>A = 5π(2)<sup>2</sup> + 5(4)(6) = 182.83</P>
<P>Divide this by 25 and round up to the next can: 182.83 / 25 = 7.3...</P>
<P>8 cans of paint are needed.</P>
<P>If you don't round up, you will not have enough paint to finish the job.</P>
<BR><BR><BR><P><strong>29.</strong> <em>Using a compass and straightedge, construct a midsegment of △AHL below.
[Leave all construction marks.]
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</em>
<P><b>Answer: </b>
<BR>To construct a midsegment, you need two midpoints. To get two midpoints, you need to construct two perpendicular bisectors. You have a choice of which side you want to bisect. You may NOT use a ruler to measure the midpoints.
<P>From point H, make an arc that is more than half the length of HL and swing it across HL and HA. (You could make two separate arcs, but there's no reason not to use the same one.</P>
<P>From point A, make an arc that intersects the first arc in two places. From point L, construct a third arc that intersect the arc drawm from point H.</P>
<P>Draw the two perpendicular bisectors. These will give you the midpoints. Use a straightedge to connect the midpoints. This is the midsegment, which is half the size of AL and parallel to AL.</P>
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<BR><BR><BR><P><strong>30.</strong> <em>
Right triangle STR is shown below, with m∠T = 90°. Altitude TQ is drawn to SQR, and TQ = 8.
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<br>If the ratio SQ:QR is 1:4, determine and state the length of SR.
</em>
<P><b>Answer: </b>
<BR>Label SQ x and QR 4x. The product of (SQ)(QR) is equal to the square of (QT) by the Right Triangle Altitude Theorem.
<CENTER>
<P>(x)(4x) = 8<sup>2</sup></P><P>4x<sup>2</sup> = 64</P><P>x<sup>2</sup> = 16</P><P>x = 4</P></CENTER>
<P>SR is x + 4x = 5x, and x = 4, so 5x = 5(4) = 20. SR = 20.</P>
<BR><BR><BR><P><strong>31.</strong> <em> Line AB is dilated by a scale factor of 2 centered at point A.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGyFVz3izACEkReqim68NLPWfpPkJVZjiXjC7YxpQqrnMm3lzGFiGq9O-RNUNpJ60GEDUkEDaz60_GrNinC9MUOca2PNqSKITXGz23BLcpw8_WqkZLD7KyIvydttSY-gn4pgK4aEe-6ImSfDzQ7CVx84i9rh01-merSLD3Aao4ic6qWlz29zBMYw/s1600/temp00.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" data-original-height="114" data-original-width="326" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGyFVz3izACEkReqim68NLPWfpPkJVZjiXjC7YxpQqrnMm3lzGFiGq9O-RNUNpJ60GEDUkEDaz60_GrNinC9MUOca2PNqSKITXGz23BLcpw8_WqkZLD7KyIvydttSY-gn4pgK4aEe-6ImSfDzQ7CVx84i9rh01-merSLD3Aao4ic6qWlz29zBMYw/s1600/temp00.png"/></a></div>
<P>Evan thinks that the dilation of AB will result in a line parallel to AB, not passing through points
A or B.
<P>Nathan thinks that the dilation of AB will result in the same line, AB.
<P>Who is correct?
<P>Explain why.
</em>
<P><b>Answer: </b>
<BR>Nathan is correct because every point on line AB will move twice as far from A in the same direction along the line AB. Also, since the image of point A will conincide with point A, parallel lines aren't possible.cimal because it's infinite.</P>
<P><B>End of Part II</b>
<P>How did you do?
<P>Questions, comments and corrections welcome.
<BR><BR>
<center> <table width=90% border=5 cellspacing=5 cellpadding=5>
<tr><td><h3>I also write Fiction!</h3><BR>You can now order my newest book <i><B>Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend</B></i>, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
<BR>Order the softcover or ebook at <a href="https://www.amazon.com/dp/B0CQGSFJ26">Amazon</a>.
</td>
<td> <a href="https://www.amazon.com/dp/B0CQGSFJ26"><img src=https://m.media-amazon.com/images/I/81yaN-b6DyL._SY466_.jpg height=200></a></td>
</tr>
<tr><td>
<BR>Also, check out <i><B>In A Flash 2020</B></i>, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
<BR>Available in softcover or ebook at <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/">Amazon</a>.
<P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<!-- <P>If you enjoy it, please consider leaving a rating or review on Amazon or on <a href="https://www.goodreads.com/book/show/54511947-in-a-flash-2020?from_search=true&from_srp=true&qid=n09Ycrxm1m&rank=1">Good Reads</a>.
<P>Thank you. -->
</td>
<td> <a href="https://www.amazon.com/Devilish-Divine-John-L-French-ebook/dp/B098NRRVLY"><img src=https://m.media-amazon.com/images/I/51QabBDTvJS.jpg height=200></a></td>
<td> <a href="https://www.amazon.com/Flash-2020-Christopher-J-Burke-ebook/dp/B08CWQTYBR/"><img src=https://m.media-amazon.com/images/I/51+4JVxWDxL.jpg height=200></a></td>
</tr></table></center>
<BR><BR>
</font>(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.com0