## Tuesday, February 28, 2023

### Algebra Problems of the Day (Algebra 1 Regents, January 2023)

The following questions appeared on the January 2023 Algebra 1 Regents Exam

More Regents problems.

### Algebra 1 Regents, January 2023

1. When the expression 2x(x − 4) − 3(x + 5) is written in simplest form, the result is

1) 2x2 − 11x − 15
2) 2x2 − 11x + 5
3) 2x2 − 3x − 19
4) 2x2 − 3x + 1

Answer: 1) 2x2 − 11x − 15

Multiply and then combine like terms.

2x(x − 4) − 3(x + 5)
2x2 − 8x − 3x - 15
2x2 − 11x - 15

Choice (1) is the correct answer.

2. The point (3,w) is on the graph of y = 2x + 7. What is the value of w?

1) -2
2) -4
3) 10
4) 13

What is the value of y when x = 3?

2(3) + 7 = 6 + 7 = 13, which is Choice (4).

3. Students were asked to write 2x3 + 3x + 4x2 + 1 in standard form. Four student responses are shown below.

Alexa: 4x2 + 3x + 2x3 + 1
Carol: 2x3 + 3x + 4x2 + 1
Ryan: 2x3 + 4x2 + 3x + 1
Eric: 1 + 2x3 + 3x + 4x2

Which student’s response is correct?

1) Alexa
2) Carol
3) Ryan
4) Eric

Standard for of a polynomial is that the leading term has the highest exponent, the second term has the next highest exponent, and so forth. The coefficients are not relevant in finding the leading term.

Ryan has the exponents in descending order. This is Choice (3).

4. Given f(x) = −3x2 + 10, what is the value of f(−2)?

1) -26
2) -2
3) 22
4) 46

Substitute x = -2 and evaluate. Remember Order of Operations.

f(-2) = 3(-2)2 + 10 = 3(4) + 10 = 22.

This is Choice (3).

5. Which relation is a function?

In a function, there can only be one y-value for every x-value. There can only be one output for any input. On a graph, this gives us the Vertical Line Test.

In Choice (1), none of the x-values repeat. This is the correct choice.

In Choice (2), the input 7 has two different outputs, so it is not a function.

In Choice (3), the graph fails the vertical line test. A vertical line, such as the y-axis, can be drawn through the circle and touch it in more than one place.

In Choice (4), the relation map shows that 6 in the domain maps to both 5 and 7 in the range. This means that it is not a function.

6. What is the value of the third quartile in the box plot shown below?

1) 18
2) 22
3) 36
4) 46

The Five Number Summary that creates the Box and Whisker Plot is made up of (in order): the minimum, the first quartile, the median, the third quartile, and the maximum.

The right edge of the box is the third quartile, and it has a value of 36, which is Choice (3).

7. What is the solution to 2 + 3(2a + 1) = 3(a + 2)?

1) 1/7
2) 1/3
3) -3/7
4) -1/3

Isolate the variable. Get all the a terms on the left and all the numbers on the right.

2 + 3(2a + 1) = 3(a + 2)
2 + 6a + 3 = 3a + 6
3a = 1
a = 1/3

This is Choice (2).

8. One Saturday afternoon, three friends decided to keep track of the number of text messages they received each hour from 8 a.m. to noon.
The results are shown below.

Emily said that the number of messages she received increased by 8 each hour.
Jessica said that the number of messages she received doubled every hour.
Chris said that he received 3 messages the first hour, 10 the second hour, none the third hour, and 15 the last hour.

Which of the friends’ responses best classifies the number of messages they received each hour as a linear function?

1) Emily, only
2) Jessica, only
3) Emily and Chris
4) Jessica and Chris

A linear function has a constant rate of change.

- Emily's messages increased by the same about each hour.

- Jessica's messages doubled, which is exponential, not linear.

- Chris's messages do not change at a constant rate, so it is not linear.

So the answer is Choice (1).

More to come. Comments and questions welcome.

More Regents problems.

### I also write Fiction!

You can now order 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.
Order 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.  ### Crop Circle

(Click on the comic if you can't see the full image.)
(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

No cardboard or string.

The truth is out there.

### I also write Fiction!

You can now order 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.
Order 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. ## Sunday, February 26, 2023

### Math Bait

(Click on the comic if you can't see the full image.)
(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Everyone else has already left MathBook for the next Math big thing!

These fake math "click bait" articles were things that I came up with on Twitter in response to a challenge to create one. I created two, and would've done more daily but I got distracted by the something else on my To Do list.

I tried taking an informal poll of Twitter between Mather, MathSpace and MathBook, but the more I thought of it, I thought that MathSpace should be the dying one and MathBook should be what everyone's parents use. And both Mike and Ken are parents, so there's that.

Those platforms will probably get mentioned again. I can't ignore social media. Even if it mostly ignore me. (sniff).

### I also write Fiction!

You can now order 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.
Order 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, February 24, 2023

### January 2023 Geometry Regents, Part IV

This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

### January 2023 Geometry, Part IV

A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

35. Given: Triangle DUC with coordinates D(-3,-1), U(-1,8), and C(8,6)

Prove: ∆DUC is a right triangle
[The use of the set of axes on the next page is optional.]

Point U is reflected over DC to locate its image point, U', forming quadrilateral DUCU9. Prove quadrilateral DUCU' is a square.

To show that DUC is a right triangle, find the slopes of all three sides. Two of them will be negative reciprocals, which means that the lines are perpendicular, and create a right angle.

To show that the quadrilateral is a square, find the slopes of the new lines to show that the figure is a parallelogram. The right angle makes it a rectangle. Find the length of of two consecutive sides to show that it is a square.

Slope of DC = (6 - -1)/(8 - -3) = 7/11

Slope of DU = (8 - -1)/(-1 - -3) = 9/2

Slope of UC = (6 - 8)/(8 - -1) = -2/9. DU is perpendicular to UC.

Therefore DUC is a right triangle.

Graph the three points to find U'.

What did I do in this image?

I drew a line from U that was perpendicular to CD. Since the slope fo CD is 7/11, then the slope of UU' must be -11/7. I plotted a point that was 11 units down and 7 units to the right and drew the line.

Now look at the distance from U to CD. You can see that it is half the distance from U to the new point. So this new point is U'. If you don't believe me, you can use the distance formula.

U' is located at (6,-3).

Now draw the quadrilateral DUCU'. Find the slopes of DU' and CU'.

Slope of DU' = (-3 - -1) / (6 - -3) = -2/9. DU' is parallel to UC.

Slope of CU' = (-3 - 6) / (6 - 8) = -9/-2 = 9/2. CU' is parallel to DU.

DUCU' is a parallelogram. Since UC is perpendicular to DU, DUCU' is a rectangle.

Show that DU = UC and you are done.

Length of DU = √(22 + 92) = √(85).

Length of UC = √(92 + 22) = √(85).

DU = UC so consecutive sides of a rectangle are congruent so the rectangle is a square.

Alternate method: If the rectangle is a square then the diagonals are congruent and perpendicular. We already showed that they are perpendicular, with slopes 7/11 and -11/7. You only have to find the lengths of the diagonals.

Length of DC = √(112 + 72) =√(170).

Length of UU' = √(72 + 112) =√(170).

The diagonals are perpendicular and congruent so the rectangle is a square.

End of Part Eaxm

How did you do?

### I also write Fiction!

You can now order 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.  ### January 2023 Algebra 1 Regents, Part IV

This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

### January 2023

Part IV: A correct answer will receive 6 credits. Partial credit can be earned.

37. Aidan and his sister Ella are having a race. Aidan runs at a rate of 10 feet per second. Ella runs at a rate of 6 feet per second. Since Ella is younger, Aidan is letting her begin 30 feet ahead of the starting line.

Let y represent the distance from the starting line and x represent the time elapsed, in seconds.

Write an equation to model the distance Aidan traveled.

Write an equation to model the distance Ella traveled.

On the set of axes below, graph your equations.

Exactly how many seconds does it take Aidan to catch up to Ella? Justify your answer.

Take each question one at a time. Any mistakes made can only be penalized once if you carry the error through to the end.

Aiden runs at 10 ft/sec, so the distance he travels is y = 10x.

Ella runs at 6 ft/sec, but has a 30 ft head start, so the distance she travels is y = 6x + 30.

The graph will look like the one below. Make sure you label at least one line. You'll notice that the lines do not intersect at wrong numbers. Don't get thrown off by that.

Since the lines cross between 6 and 7, it's better to solve it algebraically.

10x = 6x + 30
4x = 30
x = 7.5

At x = 7.5 seconds, Aiden ran 10(7.5) = 75 feet and Ella ran 6(7.5) + 30 = 75 feet.

End of Part Exam

How did you do?

More Regents problems.

### I also write Fiction!

You can now order 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, February 23, 2023

### January 2023 Geometry Regents, Part III

This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

### January 2023 Geometry, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

32. Sally and Mary both get ice cream from an ice cream truck. Sally’s ice cream is served as a cylinder with a diameter of 4 cm and a total height of 8 cm. Mary’s ice cream is served as a cone with a diameter of 7 cm and a total height of 12.5 cm. Assume that ice cream fills Sally’s cylinder and Mary’s cone.

Who was served more ice cream, Sally or Mary? Justify your answer.

Determine and state how much more is served in the larger ice cream than the smaller ice cream, to the nearest cubic centimeter.

Calculate the volume of the cylinder and the volume of the cone. Compare them and then subtract to get the second answer. Make sure to use the radius and not the diameter.

Volume of cylinder: π (2)2(8) = 100.53

Volume of cone: (1/3) π (3.5)2(12.5) = 160.35

Difference: 160.35 - 100.53 = 59.82

Mary got 60 cc more ice cream.

You must state the name or you will lose a credit for not answering the question.

33. Given: ∆AEB and ∆DFC, ABCD, AE || DF, EB || FC, AC ≅ DB

Prove: ∆EAB ≅ ∆FDC

The parallel lines and transversals give you alternate interior and exterior angles. The fact that AC = BD gives you the fact that says AB and CD must be congruent after using the Reflexive Property and the subtraction property.

 Statement Reason 1. ∆AEB and ∆DFC, ABCD, AE || DF, EB || FC, AC ≅ DB Given 2. ∠A ≅ ∠D Alternate Interior Angles 3. ∠ABE ≅ ∠DCF Alternate Exterior Angles 4. BC ≅ BC Reflexive Property 5. AB ≅ CD Subtraction Postulate 6. ∆EAB ≅ ∆FDC ASA

34. Barry wants to find the height of a tree that is modeled in the diagram below, where ∠C is a right angle. The angle of elevation from point A on the ground to the top of the tree, H, is 40°.
The angle of elevation from point B on the ground to the top of the tree, H, is 80°. The distance between points A and B is 85 feet.

Barry claims that ∆ABH is isosceles. Explain why Barry is correct.

Determine and state, to the nearest foot, the height of the tree.

You can't talk and "explain" your way out of this without backing it up with some data, some facts. All that a paragraph of text will get you is the chance to say something incorrect.

Angle HBC is 80 degrees, so Angle ABH is 100 degrees. This means that angle AHB is 40 degrees.

Note that the Exterior Angle Theorem tells us that AHB = 80 - 40, which is 40 degrees.

Since Angle A and Angle AHB are congruent, the triangle is isosceles.

Since ABH is isosceles, then HB is 85 feet. This means we can use the sine ratio to find the height of the tree.

sin 80 = x / 85
x = 85 * sin 80 = 83.7

The tree is 84 feet tall.

End of Part III

How did you do?

### I also write Fiction!

You can now order 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.  ### January 2023 Algebra 1 Regents, Part III

This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

### January 2023

Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One computional or graphing mistake will lose 1 point. A conceptual error will lose 2 credits.

33. Anessa is studying the changes in population in a town. The graph below shows the population over 50 years.

State the entire interval during which the population remained constant.

State the maximum population of the town over the 50-year period.

Determine the average rate of change from year 30 to year 40.

Explain what your average rate of change means from year 30 to year 40 in the context of the problem.

Each question is worth one credit, basically.

The population remained constant from year 20 to year 30.

The maximum population was 10,000 (during years 20 to 30).

The average rate of change from year 30 to 40 was -6000/10 or -600 people per year.

The town lost 600 people per year from year 30 to year 40.

34. The table below shows the number of math classes missed during a school year for nine students, and their final exam scores.

Write the linear regression equation for this data set. Round all values to the nearest hundredth.

State what the correlation coefficient indicates about the linear fit of the data.

Put all the data in your graphing calculator and run a linear regression. Make sure you have DiagnosticsOn.

The equation, when rounded, is y = -2.81x + 97.55

The correlation coefficient is r = -.97.

The correlation coefficient indicates that there is a strong, negative relationship.

35. A fence was installed around the edge of a rectangular garden. The length, l, of the fence was 5 feet less than 3 times its width, w. The amount of fencing used was 90 feet.

Write a system of equations or write an equation using one variable that models this situation.

Determine algebraically the dimensions, in feet, of the garden.

If the length if 5 less than 3 times the width, then l = 3w - 5. That's just translating the sentence.

The perimeter of a rectangular fence is P = 2L + 2W.

2(3w - 5) + 2w = 90

2(3w - 5) + 2w = 90
6w - 10 + 2w = 90
8w = 100
w = 12.5

The width is 12.5 and the length is 3(12.5) - 5 = 32.5.

Again, don't let fractions and decimals throw you off.

36. Given:
3y - 9 < 12
y < -2x - 4

Graph the system of inequalities on the set of axes below.

State the coordinates of a point that satisfies both inequalities. Justify your answer.

Notice that the first inequality DOES NOT have an x term. You will graph a horizontal line after you rewrite it to isolate y.

3y - 9 < 12
3y < 21
y < 7

When graphing inequalities, remember that "less than or equal to" gets a solid line, and "less than" gets a broken or dashed line. Any point on the dashed line is NOT part of the solution. Don't pick any of those points for the final part of the question, especially not the point where the two lines meet.

Your final graph should look like the one beow. Make sure you label at least one of the lines. And add an "S" to the area that is the solution.

One point that satisfies both conditions is (-10,0) because it's in the double-shaded section of the graph.

You can also "justify" your answer by plugging it into both inequalites and showing that they are true statements.

End of Part III

How did you do?

More to come. Comments and questions welcome.

More Regents problems.

### I also write Fiction!

You can now order 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, February 22, 2023

### January 2023 Geometry Regents, Part II

This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

### January 2023 Geometry, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

25. Using a compass and straightedge, construct the angle bisector of /ABC. [Leave all construction marks.]

Angle cisectors are simple. The first thing you want to do is make sure that you start at point B.

To bisect an angle, put the compass at point B and make an arc that crosses AB and BC. Then without changing the size of the compass, go to the intersections on AB and BC and make another arc in the middle of the triangle. The point where these two arcs intersect will lie on the angle bisector. Use your straightedge to draw the bisector from B through this line.

26. On the set of axes below, ∆ABC and ∆DEF are graphed.

Describe a sequence of rigid motions that would map ∆ABC onto ∆DEF.

A rotation of 90 degrees clockwise centered on the origin, followed by a translation of 1 unit to the right and four units down would map ABC onto DEF.

You don't need to write an equation, but you could. If you did, make sure you write it in the correct order.

There are other possibilities, depending upon where you center the rotation. You MUST mention the center of the rotation. You can ASSUME it's the origin. The person scoring you will not.

27. As shown in the diagram below, a symmetrical roof frame rises 4 feet above a house and has a width of 24 feet.

Determine and state, to the nearest degree, the angle of elevation of the roof frame.

Draw the altitude of the triangle. Now you have two right triangles with legs of 4 and 12.

The angle of elevation is the bottom angle and it can be found using the tangent ratio.

tan x = 4/12, so x = tan-1(4/12) = 18.4, which is 18 degrees.

28. Directed line segment AB has endpoints whose coordinates are A(-2,5) and B(8,-1). Determine and state the coordinates of P, the point which divides the segment in the ratio 3:2. [The use of the set of axes below is optional.]

Segmenting (or partitioning) AB into a ratio of 3:2 means that you need two cut it into five pieces. You want AP to be 3 of those pieces long and PB to be 2 of them.

(8 - -2)/5 = 2 and (-1 - 5)/5 = -6/5 = -1.2

2 * 3 = 6 and -1.2 * 3 = -3.6

P is at (-2 + 6, 5 - 3.6), which (4, 1.4).

Don't let the decimal throw you off. Decimals get used in real life, too.

29. In ∆ABC, AB = 5 5, AC 5 12, and m∠A = 90°. In ∆DEF, m∠D = 90°, DF = 12, and EF = 13. Brett claims ∆ABC ≅ ∆DEF and ∆ABC ~ ∆DEF. Is Brett correct? Explain why.

Both triangles are right triangles. Both triangles have two sides listed. You can use Pythagorean Theorem to show that both triangles are 5-12-13 triangles. Or you can pretty much just state that because they are common triangles, used all the time.

Since the two triangles are have congruent corresponding sides then by SSS the triangles are congruent. If two triangles are congruent, they must also be similar.

30. The volume of a triangular prism is 70 in3. The base of the prism is a right triangle with one leg whose measure is 5 inches. If the height of the prism is 4 inches, determine and state the length, in inches, of the other leg of the triangle.

If the prism has a height of 4 and a volume of 70, then the area of the base is 70/4 = 17.5 in2.

The area of a triangle = 1/2 (5) b = 17.5

So b = 17.5 (2) (1/5) = 7

31. Triangle ABC with coordinates A(-2,5), B(4,2), and C(-8,-1) is graphed on the set of axes below.
Determine and state the area of ∆ABC.

There isn't a shortcut. The best method is to make a rectangle around the triangle and calculate the negative space and subtract it from the area of the rectangle until only the triangle remains.

Make rectangle CDEF so that A is on DE and B is on EF.

Area of CDA = 1/2(6)(6) = 18. Area of AEB = 1/2(6)(3) = 9. Area of BFC = 1/2(12)(3)= 18.

Area of CDEF = (6)(12) = 72, and 72 - 18 - 9 - 18 = 27

End of Part II

How did you do?

### I also write Fiction!

You can now order 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.  ### January 2023 Algebra 1 Regents, Part II

This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

### January 2023

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.

25. Graph the function g(x) = √(x + 3) on the set of axes below.

Put the equation in your graphing calculator. You will notice that the doman (the x values) start at x = -3, and that the function is undefined for x < -3. Only plot the points that have integer values, and only put an arrow on the right end.

The points you should plot are (-3,0), (-2,1), (1, 2), (6,3), and (13,4). There's a reason that they put the g(x) axis so far over to the left.

Your graph should look something like this:

In this example, a = 1, so the left side of the graph will have a slope of -1 and the right side will have a slope of 1. (h,k) is (-1,0) because there is a plus sign instead of a minus, and there is no k.

The graph will look like this:

26. The sixth-grade classes at West Road Elementary School were asked to vote on the location of their class trip. The results are shown in the table below.

Determine, to the nearest percent, the percentage of girls who voted for Splashdown.

To quote something I may have heard before, which applies to this problem specifically, boys don't matter.

You are only looking at the number of girls that voted for Splashdown and the total number of girls.

That would be 46 / (39 + 46 + 37) = 0.377..., which is 38%.

27. Solve the inequality -2/3 x + 6 > -12 algebraically for x.

Isolate the variable. If you multiply or divide the inequality by a negative number, flip the sign around.

-2/3 x + 6 > -12

-2/3 x > -18

x < -18(-3/2)

x < 27

28. Determine the common difference of the arithmetic sequence in which a1 = 3 and a4 = 15.

You can find the common difference the same way you'd find rate of change between the points (1,3) and (4,15).

(15 - 3) / (4 - 1) = 12 / 3 = 4.

You could also make buckets like 3, ___, ___, 15 and figure out what has to go in-between. They're 12 apart and it's three steps from 3 to 15.

There are many ways to imagine the problem with a similar method for solving.

29. Given: A = √(363) and B = √(27)

Explain why A + B is irrational.

Explain why A * B is rational.

You don't have to get crazy wit the answers (even the irrational one! -- That's a joke.) Each question is worth 1 credit. You can keep it simple.

A + B is irrational because the sum of two irrational numbers will be irrational. You can write the sum, but make sure to add the "...". If you say anything else, don't say anything incorrect, like saying that it doesn't terminate without saying it doesn't repeat.

A * B is rational because they have a product of 99, which is a rational number.

This is true because 363 = (3)(11)(11) and 27 = (3)(3)(3), so the product will have 4 factors of (3) and two factors of (11). There is an even number of all the factors, so it is a perfect square. The square root of that product will have two factors of (3) and one factor of (11).

30. Use the quadratic formula to solve x2 - 4x + 1 = 0 for x.
Round the solutions to the nearest hundredth.

The quadratic formula was specifically requested so any other method will lose half credit, which in this case is 1 credit.

Look at the image below:

x = 3.73 or x = 0.27

Don't forget to round up. That would be a silly way to lose 1 credit on a 2 credit problem after doing all that work.

31. Factor completely: 4x3 - 49x

"Factor completely" almost always means multiple steps. The expression looks like a difference of perfect squares, but it isn't because the exponents are wrong. However, you can factor out one factor of x.

4x3 - 49x = x(4x2 - 49) = x(2x + 7)(2x - 7)

A difference of perfect squares will factor into two conjugates.

32. The function g is defined as

g(x) = |x + 3|, x < -2
g(x) = x2 + 1, -2 < x < -2
On the set of axes below, graph g(x).

Plot the first expression for values of x that are less than -2, and plot the second expression only when x is between -2 and 2, inclusive.

It should look like this:

Some things to note: The absolute value graph must end with an OPEN CIRCLE because it is a boundary and it is not a part of the graph of the solution. The left side should have an arrow because the domain goes to negative infinity. There should be NO ARROWS on the second part because it is a specific range with a starting and an ending point. Both of those should be closed circles, without arrows.

End of Part II

How did you do?

More to come. Comments and questions welcome.

More Regents problems.

### I also write Fiction!

You can now order 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, February 18, 2023

### Rotate

(Click on the comic if you can't see the full image.)
(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

And do it often. As recommended.

We're in the Transformations chapter of Geometry right now, so these are coming to mind more readily. Plus they are easier to create than some of the more involved or elaborate jokes waiting in the queue.

P>Dated yesterday because it was supposed to have been posted after my book club meeting. It was not.

Opinions always welcome. Spam is not.

### I also write Fiction!

You can now order 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.
Order 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, February 14, 2023

### Dilate

(Click on the comic if you can't see the full image.)
(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Better than Die Now!

They are both dilations. However, when the scale factor is less than 1, the image is reduced in size.

Happy Valentines Day!

It wasn't my plane to stop celebrating specific days, but I seem to be lucky to get in the comics that I plan without having to plan some extra ones for special days like today and the Super Bowl. Oddly enough, or evenly enough, when I kept to a regular schedule, I was forced to observe these if only as filler, but sometimes I got good jokes that way. Now I'd just like some more time. (My commute is a bit of a killer, even when I'm not out walking for an hour or two.)

### I also write Fiction!

You can now order 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.
Order 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. 