Monday, May 23, 2022

This One's For the Birds

(Click on the comic if you can't see the full image.)
(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Yes, I used that pun in the Title.

There are two pictures because one good tern deserves another.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2011)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2011

Part III: Each correct answer will receive 4 credits. Partial credit is available


36. Write the binomial expansion of (2x − 1)5 as a polynomial in simplest form.

Answer:


The long way is to write out (2x - 1)(2x - 1)(2x - 1)(2x - 1)(2x - 1) and multiply. That's a long way to go for the points, but you'll get full credit.

Binomial expansion follows Pascal's Triangle. The 5th row of Pascal's Triangle is

1 5 10 10 5 1

You can find those numbers making the triangle or using 5CN. (Honestly, you should know 1, 5, ..., 5, 1. You should only have to check the middle numbers if you didn't know them.)

So the binomial expansion is:

(1)(2x)5(-1)0 + (5)(2x)4(-1)1 + (10)(2x)3(-1)2 + (10)(2x)2(-1)3 + (5)(2x)1(-1)5 + (1)(2x)0(-1)6

32x5 - 80x4 + 80x3 - 40x2 + 10x - 1.

The (-1)n changes the signs from plus to minus. An exponent of 0 means that that factor is 1.





37. In △ABC, m∠A = 32, a = 12, and b = 10. Find the measures of the missing angles and side of △ABC. Round each measure to the nearest tenth.

Answer:


Use the law of Sines to find the measure of ∠B. Subtract A and B from 180 to find m∠C. Use the law of Sines again to find the length of side c.

Sin A / a = Sin B / b

Sin 32 / 12 = Sin B / 10

Sin B = 10 (Sin 32) / 12

Sin B = 0.4416

B = 26.2 degrees




m∠C = 180 - (32 + 26.2) = 121.8




Sin A / a = Sin C / c

Sin 32 / 12 = Sin 121.8 / c

c = 12 (Sin 121.8) / Sin 32

c = 19.24 = 19.2





38. The probability that the Stormville Sluggers will win a baseball game is 2/3. Determine the probability, to the nearest thousandth, that the Stormville Sluggers will win at least 6 of their next 8 games.

Answer:


The probability that they will win at least 6 games is the probability that they will win six games plus the probability that they will win seven games plus the probability that they will win eight games.

P(6 wins) = 8C6(2/3)6(1/3)2 = 0.273129...

P(7 wins) = 8C7(2/3)7(1/3)1 = 0.1560737...

P(8 wins) = 8C8(2/3)8(1/3)0 = 0.039018...

0.27312 + 0.15607 + 0.03901 = 0.4682 = 0.468.




End of Part III.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Saturday, May 21, 2022

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2011)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2011

Part II: Each correct answer will receive 2 credits. Partial credit is available


32. The graph below represents the function y = f(x).

State the domain and the range of this function.


Answer:


The domian is the set of all x values, which run from -5 to 8, inclusive, of [-5,8].

The range is the set of all y values, which run from -3 to 2, inclusive, or [-3,2].

Note that you need square brackets because the endpoints are included.

Note that the y values are not dependent on the endpoints of the function. You want the maximum and the minumum values for y.

You could also express your answers using inequalities: -5 < x < 8 and -3 < y < 2. Again, make you use < and not <.





33.Express [see image] in simplest radical form.



Answer:


Look at the image below:

Put the entire fraction under the radical. Simplify the terms by dividing 108 by 6, subtracting 1 from 5, and subtracting 5 from 8. Next take the square root of the factors that are perfect squares. That leaves only 2y under the radical sign.





34. Assume that the ages of first-year college students are normally distributed with a mean of 19 years and standard deviation of 1 year.

To the nearest integer, find the percentage of first-year college students who are between the ages of 18 years and 20 years, inclusive.

To the nearest integer, find the percentage of first-year college students who are 20 years old or older.


Answer:


Students who are 18 years old are one standard deviation below the mean. Students who are 20 years old are one standard deviation above the mean. If you look at the diagram of the Normal Curve Standard Deviation, the percentage of students within one standard deviation from the mean is:

15.0 + 19.1 + 19.1 + 15.0 = 68.2 per cent.

68% are between 18 and 20.

The students who are 20 years old or older are more than 1 standard deviation away from the mean. According to the table, 9.2 + 4.4 + 1.7 + 0.5 + 0.1 = 15.9. So 16% of the first-year college students are 20 years old or older.





35. Starting with sin2 A + cos2 A = 1, derive the formula tan2 A + 1 = sec2 A.

Answer:


Use the identities you know.

In particular, tan = sin / cos and sec = 1 / cos.

sin2 A + cos2 A = 1

(sin2 A)/(cos2 A) + (cos2 A)/(cos2 A) = 1/(cos2 A)

tan2 A + 1 = sec2 A




End of Part II.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Friday, May 20, 2022

Solve For X

(Click on the comic if you can't see the full image.)
(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Believe it or not, not based on a true story. But it does sound like one. I'd be impressed if a student said it, actually.

The answer is let as an exercise for any students that the teachers in my audience want to inflict this on.

I hate to say that I'll be more productive when summer gets here, because that's over a month away. But with the tech issues I have at work, even when I have free time to work on something, it's too frustrating which makes it difficult to focus.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2011)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2011

Part II: Each correct answer will receive 2 credits. Partial credit is available


28. Solve algebraically for x: 162x + 3 = 64x + 2

Answer:


Make the bases the same, and then compare the exponents.

Both 16 and 64 are is a powers of 2.

Rewrite the equation as 2(4)(2x + 3) = 2(6)(x + 2)

Now solve (4)(2x + 3) = (6)(x + 2)

(4)(2x + 3) = (6)(x + 2)

8x + 12 = 6x + 12

8x = 6x

2x = 0

x = 0

You can check your work if you don't believe 0 is the answer:

163 = 4096, 642 = 4096





29.Find, to the nearest tenth of a degree, the angle whose measure is 2.5 radians

Answer:


There are 180 degrees in π radians. Set up a proportion and solve it:

180/π = x/2.5

πx = (2.5)(180)

x = (2.5)(180)/π

x = 143.23...

Approximate 143.2 degrees to the nearest tenth of a degree.

Since 180 degrees is approximately 3.141592... radians, this seems like a reasonable amount.

If you calculated the number of radians in 2.5 degrees (correctly), you would receive one credit.





30. For a given set of rectangles, the length is inversely proportional to the width. In one of these rectangles, the length is 12 and the width is 6. For this set of rectangles, calculate the width of a rectangle whose length is 9.

Answer:


Inversely proportional means that LW is constant, so L1W1 = L2W2

(12)(6) = (9)W2

W2 = 72/9 = 8





31. Evaluate


Answer:


You can do this the long way or the short way.

The long way:

n = 1, (13 - 1) = 1 - 1 = 0

n = 2, (23 - 1) = 8 - 1 = 7

n = 3, (33 - 1) = 27 - 1 = 26

n = 4, (43 - 1) = 64 - 1 = 63

n = 5, (53 - 1) = 125 - 1 = 124

10 + 0 + 7 + 26 + 63 + 124 = 230

Quicker would be the know that 13 + 23 + 33 + 43 + 53 = (1 + 2 + 3 + 4 + 5)2.

So 10 + 152 - 5 = 10 + 225 - 5 = 230.




More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Thursday, May 19, 2022

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2011)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2011

Part I: Each correct answer will receive 2 credits.


25. A four-digit serial number is to be created from the digits 0 through 9. How many of these serial numbers can be created if 0 can not be the first digit, no digit may be repeated, and the last digit must be 5?

1) 448
2) 504
3) 2240
4) 2520

Answer: 1) 448


There are 8 choices for the first digit (not 0 and not 5).

There are 8 choices for the second digit (not 5 and not the 1st digit).

There are 7 choices for the third digit (not 5 and neither of the 1st 2 digits).

There's 1 choice for the fourth digit (5).

8 * 8 * 7 * 1 = 448, which is Choice (1).

Since the fourth digit had to be 5, only three digits were in play, which meant there were less than 1000 possibilities. So Choices (3) and (4) could be eliminated immediately.





26. Which equation represents the circle shown in the graph below that passes through the point (0,−1)?

1) (x − 3)2 + (y + 4)2 = 16
2) (x − 3)2 + (y + 4)2 = 18
3) (x + 3)2 + (y - 4)2 = 16
4) (x + 3)2 + (y - 4)2 = 18

Answer: 2) (x − 3)2 + (y + 4)2 = 18


The equation of a circle is (x − h)2 + (y - k)2 = r2, where (h,k) is the center of the circle and r is the radius.

Notice that the radius is a little more than 4, so r2 is not 16. The center is at (3,-4) and there is a point on the circle at (0,-1). You can draw a right triangle with legs equal to 3 and 3. That would mean that the hypotenuse, which is the radius of the circle, has a length of √(18). That means that r2 = 18. Eliminate Choices (1) and (3).

There are minus signs in the formula, so the numbers are flipped. Choice (2) is correct.





27. Which task is not a component of an observational study?

1) The researcher decides who will make up the sample
2) The researcher analyzes the data received from the sample.
3) The researcher gathers data from the sample, using surveys or taking measurements.
4) The researcher divides the sample into two groups, with one group acting as a control group.

Answer: 4) The researcher divides the sample into two groups, with one group acting as a control group.


An observational study does not have a control group.




End of Part I.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Wednesday, May 18, 2022

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2011)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2011

Part I: Each correct answer will receive 2 credits.


21. For which equation does the sum of the roots equal −3 and the product of the roots equal 2?

1) x2 + 2x - 3 = 0
2) x2 - 3x - 2 = 0
3) 2x2 + 6x + 4 = 0
4) 2x2 - 6x + 4 = 0

Answer: 3) 2x2 + 6x + 4 = 0


The product of the roots is C/A, and the sum is -B/A.

Choice (1) has a sum of -3/1 = -3. Eliminate Choice (1).

Choice (2) has a sum of -2/1 = -2. Eliminate Choice (2).

Choice (3) has a sum of 4/2 = 2, and a product of -6/2 = -3. This is the correct choice.

Choice (4) has a sum of 4/2 = 2, and a product of -(-6)/2 = 3. Eliminate Choice (4).





22. The expression (2x + 4) / (√(x + 2)) is equivalent to


Answer: 4) [See Image]


See image.

Multiply by &radic(x+2)/&radic(x+2). Then factor (2x + 4)/(x + 2) = 2.





23. Which equation is sketched in the diagram below?



1) y = csc x
2) y = sec x
3) y = cot x
4) y = tan x

Answer: 1) y = csc x


The U shaped graphs are secant and cosecant, not tangent and cotangent. Eliminate Choices (3) and (4).

Sec x = 1 / cos x and csc x = 1 / sin x.

At x = 0, cos x = 1 and sin x = 0. That means that csc x is undefined at 0, as shown in the graph.

The correct choice is (1).





24. The expression log5(1/25) is equivalent to

1) 1/2
2) 2
3) -1/2
4) -2

Answer: 4) -2


If x = log5(1/25) then 5x = 1/25.

So x = -2, which is Choice (4).




More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Friday, May 13, 2022

The Square

(Click on the comic if you can't see the full image.)
(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

When now we know who's the kite in the breeze...

I liked the idea behind this comic and I was really getting into it when I was coming up with dialogue ... until I remembered that it's a four- (or six-) paneled strip and I didn't have room for all that. And, let's face it, no one want to read a novel like this.

The crux of it is that Bibi was friends with each of the other two first, and they were friends for different reasons. So Freedom and Serenity (Ningjing) were friends through Bibi but weren't originally friends with each other. (Granted, since their first appearance, they've always been in the same strips. However, I was creating backstory.)

I also contemplated having Mr. Wayne appear to make a comment about Geometry class. This exercise is left to the imagination of the reader.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Wednesday, May 11, 2022

Cellphone

(Click on the comic if you can't see the full image.)
(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Except when two lines are on top of one another...

I usually refreain from commenting on people's dumb arguments, whether I agree with them or not. (Yes, there are quite a few out there who will argue positions I may agree with, but they argue them very badly. Heaven forbid that you should try to correct them to make their argument stronger.)

A problem with social media (I almost wrote "the problem") is that too many people seek the perfect applause line or "mic drop". Then in creating such a construction (using a process similar to that of Dr. Frankenstein) they eschew any sort of logic without taking into account if their position is upon solid purchase (as opposed to shaky ground, don't you know).

Some responses are so smug that the creators don't realize that there isn't anything clever about them at all and can usually be shot down with a single retort. This is generally taken as an inverted measure of their cleverness.

Social media is a crazy place.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Wednesday, May 04, 2022

School Life #27: Ma the Force

(Click on the comic if you can't see the full image.)
(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Ma the Force move you ... or something like that.

And when someone gives me the May the Fourth be with you, I always return "And also with you."



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Friday, April 29, 2022

One-Off Topics

(Click on the comic if you can't see the full image.)
(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Like the compass and straightedge you pull out for that one lesson before you complain about how lousy the compasses are these days.

Inspiration for this came from my cropping a group of students from a "characters" file to include in the backgound of a slide in one of my classes. Then it only made sense to include the perennial substitute, Quinn Jonas.

The tricky parts were deciding what topic to mention (before realizing that I didn't actually have to mention one), and what I was going to title this strip.

Speaking of titles and names of things: any suggestions for these guys? The one with the X across his shirt might have a name, although possibly not canonical.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Tuesday, April 26, 2022

Algebra Problems of the Day (Integrated Algebra Regents, January 2011)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones. The Integrated Algebra Regents covered most of the same material as the current Algebra Regents, with a few differences.

More Regents problems.

Integrated Algebra Regents, January 2011

Part IV: Each correct answer will receive 4 credits. Partial credit is available.


37. An oil company distributes oil in a metal can shaped like a cylinder that has an actual radius of 5.1 cm and a height of 15.1 cm. A worker incorrectly measured the radius as 5 cm and the height as 15 cm.

Determine the relative error in calculating the surface area, to the nearest thousandth.

Answer:


Remember that they are asking for relative error, which is a decimal, not the percent of error. If you give the answer as a percentage, you will lose a point. (No, I don't agree with that, but I don't write the rubric.)

The relative error is the difference between the actual amount and the measured amount, divided by the actual amount.

You don't actually have to find the two surface areas, but you need to have the factors in your equation. Why would you do this? Because you can eliminate pi from the equation.

Surface area of a cylinder is equal to 2 π r2 + 2 π r h

( (2 π (5.1)2 + 2 π (5.1)(15.1)) - (2 π (5)2 + 2 π (5)(15) )
(2 π (5.1)2 + 2 π (5.1)(15.1))

2π( ((5.1)2 + (5.1)(15.1)) - ((5)2 + (5)(15) )
((2 π)((5.1)2 + (5.1)(15.1)))

( ((5.1)2 + (5.1)(15.1)) - ((5)2 + (5)(15) )
((5.1)2 + (5.1)(15.1))

( 103.02 - 100 )
103.02

3.02
103.02

Which is approximately equal to 0.029 to the nearest thousandth.





38. The Booster Club raised $30,000 for a sports fund. No more money will be placed into the fund.
Each year the fund will decrease by 5%. Determine the amount of money, to the nearest cent, that will be left in the sports fund after 4 years.

Answer:


You can use the equation for exponential decay and get it in one shot, or you can you the calculate 5% of the total, subtract it and then repeat that three more times. You will get the same answer.

Do the long one if you don't remember how to do the quick method.

A = 30000(1.00 - 0.05)4 = 24435.19, to the nearest cent.

The longer way:

100 - 5% = 95% that remains, which is .95.

30000 * 0.95 = 28500

28500 * 0.95 = 27075

27075 * 0.95 = 25721.25

25721.25 * 0.95 = 24435.1875 = 24435.19

Note: According to the Answer Key, a result of 24435.20 was also acceptable because different models and brands of calculators yield slightly different results.





39. Graph the following system of inequalities on the set of axes shown below and label the solution set S.

y > −x + 2
y ≤ 2/3 x + 5


Answer:


Thankfully, the inequalities are already in slop-intercept form and are ready to be entered into your graphing calculator, if you have one.

Most that the first inequality says greater than. The line will be broken (dashed, dotted) and the graph will be shaded Above the line. The second inequality says less than or equal to. This line will be solid (part of the solution). The graph will be shaded below the line.

Make sure that you label the two lines, even if it looks "obvious" to you. Look at the graph below:




End of Exam.




More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Monday, April 25, 2022

Restful Sleep

(Click on the comic if you can't see the full image.)
(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

If one wakes you up, the other will keep you up. Of course, one is more easier remedied than the other.

Age would be another factor.

I almost typed "fullness" on the bottom axis until it occured to me that I could use two different meanings for "volume".



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2011)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2011

Part I: Each correct answer will receive 2 credits.


16. If x2 + 2 = 6x is solved by completing the square, an intermediate step would be

1) (x + 3)2 = 7
2) (x - 3)2 = 7
3) (x - 3)2 = 11
4) (x - 6)2 = 34

Answer: 2) (x - 3)2 = 7


First, 6x needs to be subtracted from both sides. To complete the square, you will end up with half of -6, which is -3. Eliminate Choice (1) and (4).

x2 + 2 = 6x

x2 - 6x + 2 = 0

x2 - 6x + 9 + 2 = 0 + 9

x2 - 6x + 9 = 9 - 2

(x - 32 = 7

This is Choice (2).





17. Three marbles are to be drawn at random, without replacement, from a bag containing 15 red marbles, 10 blue marbles, and 5 white marbles. Which expression can be used to calculate the probability of drawing 2 red marbles and 1 white marble from the bag?

1) 15C2 * 5C1 / 30C3
2) 15P2 * 5P1 / 30C3
3) 15C2 * 5C1 / 30P3
4) 15P2 * 5P1 / 30P3

Answer: 1) 15C2 * 5C1 / 30C3


This problem is a lot easier than it looks as soon as you see the choices. This is purely a Combination problem, not a permutation problem.

Therefore, the answer is Choice (1).





18. The expression x(-2/5) is equivalent to



Answer: 4) [See image]


The negative exponent means that the x will move to the denominator. Eliminate Choices (1) and (2).

The denominator of 5 in the exponent means that the expression is the fifth root, which is Choice (4).





19. On January 1, a share of a certain stock cost $180. Each month thereafter, the cost of a share of this stock decreased by one-third. If x represents the time, in months, and y represents the cost of the stock, in dollars, which graph best represents the cost of a share over the following 5 months?

Answer: 3) [See image]


If something drops in 1/3 of its value every month, that is exponential decay. Eliminate Choices (1) and (2) which show linear graphs.

Choice (3) shows the stock losing 1/3 of its value. This is the correct response.

Choice (4) shows the stock reduced to 1/3 of its value. It has lost 2/3.





20. In the diagram below of right triangle JTM, JT = 12, JM = 6, and m∠JMT = 90


What is the value of cot J ?

1) √(3)/3
2) 2
3) √(3)
4) 2 √(3) / 3

Answer: 1) √(3)/3


You want cotangent, which means we need the adjacent over the opposite (the inverse of tangent). We know the adjacent is 6. We don't know the opposite, but we could find it.

If you realized that this is half of an equilateral traingle, and therefore a 30-60-90 right triangle, they you know that MT is JM time %radic;(3). If you didn't realize that, you can perform Pythagorean Theorem to get the same result.

So cot J = 6 / 6 %radic;(3) = 1 / %radic;(3) = 1 / %radic;(3) * %radic;(3) / %radic;(3) = %radic;(3) / 3, which is Choice (1).




More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.