June 2022 Geometry Regents, Part 2

The following are some of the multiple questions from the recent January 2020 New York State Common Core Geometry Regents exam.

June 2022 Geometry, Part II

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

25. The Leaning Tower of Pisa in Italy is known for its slant, which occurred after its construction began. The angle of the slant is 86.03° from the ground. The low side of the tower reaches a height of 183.27 feet from the ground.

Determine and state the slant height, x, of the low side of the tower, to the nearest hundredth of afoot.

Answer:
First off, this isn't your typical Regents trigonometry ratio question, but it is still just a trigonometry question.

You have a right triangle, a leg (the height), a base angle that is opposite the leg, and you're looking for the hypotenuse. That means that you need to use sine.

sin 86.03 = 183.27 / x
x = 183.27 / sin 86.03
x = 183.7108...

To the nearest hundredth of a foot, the slant height is 183.71.

26. In the diagram below, quadrilateral ABCD is inscribed in circle 0, and mCD : mDA : mAB : mBC = 2:3:5:5. (refering to the arcs)

Determine and state m∠B.

Answer:
The sum of the four arcs is 360 degrees. Write an equation from the proportion:

2x + 3x + 5x + 5x = 360
15x = 360
x = 24

Since ∠B is an inscribed angle, it is half the size of the arc that it intercepts. It intercepts arc CDA. So ∠B = 1/2( 2(24) + 3(24)) = 60.

27. In the diagram below, a right circular cone has a diameter of 10 and a slant height of 13. Determine and state the volume of the cone, in terms of ℼ.

Answer:
The volume of a cone is 1/3 of the volume of a cylinder with the same base and height.

The radius is half of 10, which is 5. The height can be found using Pythagorean Theorem, with the radius being the other leg and the slant height being the hypotenuse of the right traingle.

I constantly remind students that you should know and recognize 3-4-5 an 5-12-13 triangles when you see them.

If you didn't, then 13² - 5² = h², so 169 - 25 = 144 = h², and h = 12.

V = 1/3 ℼ r² h = 1/3 ℼ (5)²(12) = 1/3 ℼ (25)(12) = 100ℼ

28. In the diagram below, parallelogram EFGH is mapped onto parallelogram IJKH after a reflection over line l.

Use the properties of rigid motions to explain why parallelogram EFGH is congruent to parallelogram IJKH.

Answer:
A reflection is a rigid motion where size (distance) and shape (angle measure) are preserved, so the image will be congruent to the original.

29. Izzy is making homemade clay pendants in the shape of a solid hemisphere, as modeled below. Each pendant has a radius of 2.8 cm.

How much clay, to the nearest cubic centimeter, does Izzy need to make 100 pendants?

Answer:
The volume of a hemisphere is 1/2 the volume of a sphere. So the volume is V = 1/2 (4/3 ℼ r³). You are given the radius, so there's no reason to divide it. Use the ℼ key on the calculator to get enough decimals to avoid a rounding error. DO NOT USE 3.14. (And forget about 22/7!)

V = 1/2 (4/3 ℼ r³)
V = 1/2 (4/3) (3.141592) (2.8)³
V = 45.976

The Volume for 100 pendants is 4598.

30. Determine and state the coordinates of the center and the length of the radius of the circle whose equation is x² + y² + 6x = 6y + 63.

Answer:
The equation must be put into the standard form for a circle. That means getting all the x and y terms onto the left side and then Completing the Squares.

x² + y² + 6x = 6y + 63

x² + y² + 6x - 6y = 63

x² + 6x + y² - 6y = 63

x² + 6x + 9 + y² - 6y + 9 = 63 + 9 + 9

(x + 3)² + (y - 3)² = 81

The center of the circle is (-3, 3) and the radius is √(81) = 9.

Remember to flip the signs and to take the square root of 81.

31. 1 Use a compass and straightedge to construct a line parallel to AB through point C, shown below. [Leave all construction marks.]

Answer:
To make a parallel line, draw line CA. Construct an angle congruent to CAB with C as its vertex.

End of Part II

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! 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. Thank you.

Happy Tau Day 2022

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Its proponents will always swear that its better but the general public just isn't interested.

This was something that I typed on Facebook (and possibly Twitter) six years ago. It came up today as a memory. So I shared it again. It got a couple of comments, so I made it this year's comic. (It make be evening, but it's still Tau Day!)

I've had a few different Tau characters. Most likely they'll never appear during the rest of the year. I'm not one of the proponents of tau. I teach Algebra to high school students. Let the Ultra Math Nerds use tau. (I don't count myself among their numbers even if my family and friends might.)

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.

June 2022 Algebra 1 Regents, Part IV

This exam was adminstered in June 2022. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

June 2022

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


37. At an amusement park, the cost for an adult admission is a, and for a child the cost is c. For a group of six that included two children, the cost was $325.94. For a group of five that included three children, the cost was $256.95. All ticket prices include tax.

Write a system of equations, in terms of a and c, that models this situation.

Use your system of equations to determine the exact cost of each type of ticket algebraically.

Determine the cost for a group of four that includes three children.

Answer:

If you don't get the two equations correct, the rest will be difficult. Important: Keep going. You can get partial credit for consistent work but only if you finish the problem and get an answer. Yes, the person scoring the exam will go through your work.

This is the first time that I remember a question saying "a group of six including two children" meaning that there are only 4 adults.

The equation for the first group is 4a + 2c = 325.94

The equation for the second group is 2a + 3c = 256.95

You can solve this system using elimination because 2a is a factor of 4a.

4a + 2c = 325.94
2a + 3c = 256.95

4a + 2c = 325.94
-4a - 6c = -513.90
-4c = -187.96

c = 46.99

4a + 2(46.99) = 325.94

4a + 93.98 = 325.94

4a = 231.96

a = 57.99

An adult ticket costs $57.99 and a child's ticket costs $46.99.

You can answer the last question through substitution into the expression a + 3c, or you use the amount provided in the quation for 2a + 3c and subtract c.

Method 1: 57.99 + 3(46.99) = 198.96

Method 2: 256.95 - 57.99 = 198.96

End of Exam

How did you do?

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.

Integrity of the Interval

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

I have no idea where that phrase came from. I just popped out of my mouth.

Is it something that Bro. Steve might've said? Maybe. But I'm fairly certain that he didn't. Not that I remember.

A question did come up regarding a rubric about graphing errors. If a specific interval is given, then going beyond that interval is a graphing error because it didn't follow directions. If the graphs is within the interval but has arrows at the endpoints, that implies that it continues beyond the interval. So it is an error as well.

Why are the arrows such a big deal? I stated the above without even thinking about it. I was just giving a rationale, by the way, not necessarily endorsing it. I hate when questions are set up in ways that make losing points easy. I don't mind thoughtful problems, but I don't like "gotcha" problems.

Are arrows on the endpoints a "gotcha"? The prrof is left as an exercise to the reader.

One other thing: for my newer readers, Bro. Steve appeared once before back at a time when most of the male characters were tall and thin with big feet. I found it amusing back then, but started altering the characters after a number of years.

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.

June 2022 Algebra 1 Regents, Part III

This exam was adminstered in June 2022. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

June 2022

Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One mistake computational will lose 1 point. A conceptual error will lost half credit, or 2 points. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


33. The graph below models the height of Sma's kite over a period of time.

Explain what the zeroes of the graph represent in the context of the situation.

State the time interval over which the height of the kite is increasing.

State the maximum height, in feet, that the kite reaches.

Answer:

The zeroes of the graph represent when the kite is on the ground. Its height is zero.

There are two intervals where the kite's height is only increasing and doesn't decrease.. They are 0 < t < 1/2 and 1 < t < 2.

The kites's maximum height is 60 feet, which happens at t = 2 minutes.

This problem doesn't require any work to be shown because it assumes that you are gathering information from the graph provided.

34. On the set of axes below, graph f(x) = x² - 1 and g(x) = 3^x.

Based on your graph, for how many values of x does f(x) = g(x)? Explain your reasoning.

Answer:

Graph and label both equations. Label the points of intersection.

According to the graph, there is only on value of x where f(x) = g(x), which is the point where the graphs intersect.

Note: if you make a graphing error, you should answer this question based on your graph. If your graph shows, e.g., 0 or 2 points of intersection, then you should answer that number and explain that your graph says so. You will lose credit for the graphing error but you will NOT lose credit for the final question if it is consistent with your graph.

35. An insurance agent is looking at records to be determine if there is a relationshop between a driver's age and percentage of accidents cause by speeding. The table below shows his data.

State the linear regression that models the relationshop between the driver's age, x, and the percentage of accidents caused by speeding, y. Round all values to the nearest hundredth.

State the value of the correlation coefficient to the nearest hundredth. Explain what this means in the context of the problem.

Answer:

Put the information into lists L₁ and L₂. Use the Linear regression fuction to find y = ax + b and the value of r.

To the nearest hundredth, a = -0.96, b = 64.74, and r = -0.98. (Ignore r². We don't need that.)

So y = -0.96x + 64.74 and r = -0.98.

In the context of this problem, there is a strong correlation between the age of the driver and the number of accidents caused by speeding.

36. Solve the system of inequalities graphically on the set of axes below. Label the solution set S. 2x + 3y < 9
2y > 4x + 6

Determine if the point (0,3) is a solution to this system of inequalities. Justify your answer.

Answer:

To use your graphing calculator, you need to rewrite the inequalities in slope-intercept form. Note that you must use the Original equations when labeling the graph.

2x + 3y < 9

3y < -2x + 9

y < -2/3 x + 3

2y > 4x + 6

y > 2x + 3

Remember that "<" means a dotted, dashed or broken line and you must shaded under the line. The line itself is NOT part of the solution.

The > means use a solid line and shade above the line. A solid line IS part of the solution set.

Make a large "S" in the section of the graph that is shaded twice.

The point (0,3) is not a solution because it is on the broken line and the broken line is not a part of the solution.

Note 1: If your graph is incorrect, answer according to what is on your graph if, for example, (0,3) is not on the dotted line.

Note 2: If you didn't draw the graph, you could still answer this last part algebraically for a point. Solving counts as justification. You must have justification -- "Yes" or "No" by itself will NOT get any credit.

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 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.

June 2022 Algebra 1 Regents, Part II

This exam was adminstered in June 2022. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

June 2022

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. Is the product of √(1024) and -3.4 rational or irrational? Explain your answer

Answer:

The product of √(1024) and -3.4 is -108.8, which is a rational number. All numbers with terminating decimals are rational.

If you know binary, you might have recognized that 1024 is 2¹⁰. The square root of 2¹⁰ is 2⁵, which is a rational number. The product of two rational numbers is always rational.

Note that you didn't actually have to have the product written to get full credit, so long as you stated that the radical was a rational number and the product was rational.

26. Describe the tranformations performed on the graph of f(x) = x² to obtain the graph of g(x) when g(x) = (x - 3)² - 4.

Answer:

The transformation is 3 units to the right and 4 units down.

If you weren't sure, you could have put both equations in your graphing calculator and looked at the graphs and the tables of values.

The vertex of f(x) is (0,0), and the vertex of g(x) is (3,-4).

27. The total profit earned at a garage sale during the first five hours is nodeled by the graph shown below.

Determine the average rate of change, in dollars per hour, over the interval 1 < x < 4.

Answer:

Look at the dollar values for x = 1 and x = 4. The two points are (1, 40) and (4, 100).

The average rate of change is (100 - 40) / (4 - 1) = 60 / 3 = 20

$20 per hour.

28. Subtract 3x(x - 2y) from 6(x² - xy) and express your answer as a monomial.

Answer:

A monomial is one term. This means that something in the problem is going to cancel because the terms are not alike.

When subtracting, remember that the "from" value goes FIRST.

6(x² - xy) - 3x(x - 2y)

6x² - 6xy - 3x² + 6xy

3x²

If you made one algebraic or arithmetic mistake, you could still get one credit. The word "monomial" was a big hint though.

29. A function is graphed on the set of axes below.

State the domain of the funtion.

State the range of the function.

Answer:

The domain of the function refers to all the possible x values. It is all real numbers, from negative infinity to positive infinity.

The range of the function refers to all the possible y values. It is y > 3. The range cannot be any value less than 3, which is the minimum of the function.

30. Solve 6x² + 5x - 6 = 0 algebraically for the exact values of x.

Answer:

Usually when they do to the trouble to state "exact values", it generally means that they don't want you to round any decimal solutions. This might mean the answer is irrational or that there is a repeating decimal.

If you aren't sure how to factor, you can use the Quadratic Formula.

I like using the Reverse Area model, and then Factor by Grouping. Full disclosure: I would not have said that 10 years ago. We live and learn and grown.

Using the standard form, ax² + bx + c = 0, multiply (a)(c) = (6)(-6) = -36.

What two factors of -36 have a sum of +5? A quick check of the factors of -36 will give you -9 and +4.

So 6x² + 5x - 6 = 6x² + 9x - 4x - 6

6x² + 9x - 4x - 6 = 0

3x(2x + 3x) - 2(2x + 3) = 0

(3x - 2)(2x + 3) = 0

(3x - 2) = 0 or (2x + 3) = 0

3x = 2 or 2x = -3

x = 2/3 or x = -3/2

You must write 2/3 as a fraction or use notation to indicate that it is a repeating decimal. If you write x = .7 or x = .67, etc., you would lose a point.

Quadratic Formula solution will be added soon.

31. Factor the expression x⁴ - 36x² completely.

Answer:

When you see "completely", that's a hint that there's going to be more than one step.

Factor the GCF, which is x².

x⁴ - 36x² = x²(x² - 36)

Now there is a difference of perfect squares to deal with. (You could have done this first if you wanted to.)

x²(x² - 36) = x²(x + 6)(x - 6)

32. Determine the exact values of x² - 8x - 5 = 0 by completing the square.

Answer:

If you use a different method and get the correct answer, you will only receive 1 credit. Again, "exact values" means that you shouldn't round any decimals, and should leave irrational numbers in radical form.

x² - 8x - 5 = 0

Half of -8 is -4, and (-4)² is 16, so add a zero pair of (16 - 16) to the left side of the equation.

x² - 8x + 16 - 16 - 5 = 0

Reduce the perfect square.

(x - 4)² - 21 = 0

(x - 4)² = 21

x - 4 = + √(21)

x = 4 + √(21)

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 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.

Slipped Up

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Slip ups happen all the time. Papers just move.

I wonder if this means that we'll see the same students in the same classes next year. (Ya think?)

Filling in the wrong bubbles on an exam happens, particularly when a student skips a question and forgets to leave that line blank. Sometimes the catch their error, sometimes they do not.

Personally, I hate using bubble sheets for tests if I can avoid them. I've had APs and colleagues encourage their use, and I've had large amounts of students that needed tests graded. On the other hand, in general, I hate multiple-choice questions in math class. They have some uses, but I'd rather see my students work instead of a bunch of circles.

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.

Geometry Problems of the Day (Geometry Regents, August 2010)

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

More Regents problems.

Geometry Regents, August 2010

Part I: Each correct answer will receive 2 credits.


16. In which polygon does the sum of the measures of the interior angles equal the sum of the measures of the exterior angles?

1) triangle
2) hexagon
3) octagon
4) quadrilateral

Answer: 4) quadrilateral

The sum of the external angles of ANY polygon is 360 degrees. The sum of the interior angles of a polygon is the number of sides minus 2, times 180.

A triangle has 180 degrees. A quadrilateral has 360 degrees. A pentagon has 540 degrees. A hexagon has 720 degrees. A heptagon has 900 degrees. An octagon has 1080 degrees. Etc.

The correct answer is Choice (4).

17. In the diagram below of circle O, chords AB and CD intersect at E.

If CE = 10, ED = 6, and AE = 4, what is the length of EB ?

1) 15
2) 12
3) 6.7
4) 2.4

Answer: 1) 15

If two chords intersect, the products of their segments will be equal.

So (CD)(ED) = (AE)(EB)

Which means (10)(6) = (4)(EB)

EB = 60 / 4 = 15, which is Choice (1).

18. In the diagram below of △ABC, medians AD, BE, and CF intersect at G.

If CF = 24, what is the length of FG?

1) 8
2) 10
3) 12
4) 16

Answer: 1) 8

The ratio of CF:FG is 3:1. The centroid splits to the median into two segments at a ratio of 2:1.

One third of 24 is 8, which is Choice (1).

19. If a line segment has endpoints A(3x + 5, 3y) and B(x − 1, −y), what are the coordinates of the midpoint of AB?

1) (x + 3, 2y)
2) (2x + 2, y)
3) (2x + 3, y)
4) (4x + 4, 2y)

Answer: 2) (2x + 2, y)

To find the midpoint, you have to find the middle value of the x-coordinates and of the y-coordinates. To do that, add the expressions together and divide by two.

3x + 5 + x - 1 = 4x + 4. Dividing that by two gives a quotient of 2x + 2, which is Choice (2).

3y - y = 2y. Divide that by 2, and you get just y. That is also Choice (2). (And also Choice (3), which we eliminated already.)

20. If the surface area of a sphere is represented by 144π, what is the volume in terms of π?

1) 36π
2) 48π
3) 216π
4) 288π

Answer: 4) 288π

The surface area of a sphere can be found using the formula SA = 4πr².

If 4πr² = 144π then r² = 36, and r = 6.

The Volume of a sphere can be found using the formula V = (4/3)πr³.

V = (4/3)π(6)³ = V = 288π, 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.

Algebra Problems of the Day (Integrated Algebra Regents, August 2010)

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, August 2010

Part I: Each correct answer will receive 2 credits.


11. The width of a rectangle is 3 less than twice the length, x. If the area of the rectangle is 43 square feet, which equation can be used to find the length, in feet?

1) 2x(x - 3) = 43
2) x(3 - 2x) = 43
3) 2x + 2(2x - 3) = 43
4) x(2x - 3) = 43

Answer: 4) x(2x - 3) = 43

The area of a rectangle is the length times the width. The width is 2L - 3.

So A = LW = L(2L - 3) = 43.

Substitute x for L and you have Choice (4).

12. Which value of x is the solution of (2x - 3) / (x - 4) = 2/3?

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

Answer: 2) 1/4

You can cross-multiply and solve, or you can substitute each of the choices.

(2x - 3) / (x - 4) = 2/3

3(2x - 3) = 2(x - 4)

6x - 9 = 2x - 8

4x = 1

x = 1/4

Check: (2(1/4) - 3) = -2.5, 1/4 - 4 = -3.75.
-2.5/-3.75 = 2/3. CHECK!

13. What is the perimeter of a regular pentagon with a side whose length is x + 4?
1) x² + 16
2) 4x + 16
3) 5x + 4
4) 5x + 20

Answer: 4) 5x + 20

A regular pentagon has 5 sides of equal length. If one side is x + 4, then the perimeter is 5(x + 4), or 5x + 20.

14. Which equation represents a line parallel to the y-axis?

1) x = y
2) x = 4
3) y = 4
4) y = x + 4

Answer: 2) x = 4

The y-axis has an equation of x = 0. It is parallel to all lines with equations x = a, where a is a real number.

Choice (1) x = y will intersect the y-axis when y = 0.

Choice (3) y = 4 is a horizontal line, not a vertical line. It will intersect the y-axis at (0,4).

Choice (4) y = x + 4 is not a vertical line. It will intersect the y-axis when y = 4.

15. The diagram below shows the graph of y = -x² - c.

Which diagram shows the graph pf y = x² - c?

Answer: 1) [SEE IMAGE]

A quadratic equation starting with y = -x² will create a parabola opening downward. A quadratic equation starting with y = x² will create a parabola opening upward. Eliminate Choices (2) and (3).

The y-intercept hasn't changed between the two equations. They are both at -c, where c appears to be a postivie number. Only Choice (1) has shares the same y-intercept with the given graph, so it must be the answer.

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.

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, August 2010)

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, August 2010

Part I: Each correct answer will receive 2 credits.


6. In △ABC, m∠A = 74, a = 59.2, and c = 60.3. What are the two possible values for m∠C, to the nearest tenth?

1) 73.7 and 106.3
2) 73.7 and 163.7
3) 78.3 and 101.7
4) 78.3 and 168.3

Answer: 3) 78.3 and 101.7

The Law of Sines says that Sin A / c = Sin C / c

So Sin C = c Sin A / a

Sin C = 60.3 Sin 74 / 59.2 = 0.9791...

C = 78.3, which is 90 - 11.7. The sine of (90 + 11.7) is also 0.9791.

So Choice (3) 78.3 and 101.7 are the correct answers.

7. What is the principal value of cos^-1(√(3)/2)?

1) −30°
2) 60°
3) 150°
4) 240°

Answer: 3) 150°

Where on the Unit Circle is the x value = -√(3)/2? At 150 degrees and 210 degrees.

The principal value is 150°, which is Choice (3).

8. What is the value of x in the equation 9^{3x + 1} = 27^{x + 2}?

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

Answer: 4) 4/3

If you aren't sure how to approach this, yo can always substitute and evaluate each of the choices using your calculator.

Both 9 and 27 are powers of 3, so each expression can be rewritten to have 3 as its base. If the two equal expressions have the same base then there exponents must be equal as well.

9^{3x + 1} = 27^{x + 2}

(3²)^{3x + 1} = (3³)^{x + 2}

3^{6x + 2} = 3^{3x + 6}

6x + 2 = 3x + 6

3x = 4

x = 4/3

The correct answer is Choice (4).

9. The roots of the equation 2x² + 7x - 3 = 0 are

1) -1/2 and -3
2) 1/2 and 3
3) (-7 + √(73))/4
4) (7 + √(73))/4

Answer: 3) (-7 + √(73))/4

A quick check tells you that 18 - 21 - 3 =/= 0 and 18 + 21 - 3 =/= 0. So Choices (1) and (2) are eliminated.

If you have to use the Quadratic Formula, the first thing is -b, which is -7. That is Choice (3) and not Choice (4).

10. Which ratio represents csc A in the diagram below?

1) 25/24
2) 25/7
3) 24/7
4) 7/24

Answer: 2) 25/7

The ratio csc is the inverse of sin. Since sin is OPP/HYP, then csc is HYP/OPP.

The hypotenuse is 25. The side opposites from angle A is 7. So csc A is 25/7, which is Choice (2).

Choice (1) is HYP/ADJ, which is 1/cos, which is sec.

Choice (3) is ADJ/OPP, which is 1/tan, which is cot.

Choice (4) is OPP/ADJ, which is tan.

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.

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, August 2010)

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, August 2010

Part I: Each correct answer will receive 2 credits.


1. The product of (3 + √(5)) and (3 − √(5)) is

1) 4 − 6 √(5)
2) 14 − 6 √(5)
3) 14
4) 4

Answer: 4) 4

The product of two conjugates is the Difference of the Two Squares.

(3 + √(5))(3 − √(5)) = 3² - (√(5))² = 9 - 5 = 4

2. What is the radian measure of an angle whose measure is −420°?

1) −7π/3
2) −7π/6
3) 7π/6
4) 7π/3

Answer: 1) −7π/3

The conversion rate is 180° = π radians.

-420π/180 simplifies to −7π/3, which is Choice (1)

3. What are the domain and the range of the function shown in the graph below?

1) {x | x > −4}; {y | y > 2}
2) {x | x ≥ −4}; {y | y ≥ 2}
3) {x | x > 2}; {y | y > −4}
4) {x | x ≥ 2}; {y | y ≥ −4}

Answer: 2) {x | x ≥ −4}; {y | y ≥ 2}

The domain is all the valid x values, which are x x ≥ −4. The range is all the value y values, which are y ≥ 2.

This is Choice (2).

4. The expression 2i² + 3i³ is equivalent to

1) −2 − 3i
2) 2 − 3i
3) −2 + 3i
4) 2 + 3i

Answer: 1) −2 − 3i

i² = -1 and i³ = -i.

Therefore, 2i² + 3i³ = -2 - 3i.

5. In which graph is θ coterminal with an angle of −70°?

Answer: 4) [SEE IMAGE]

The angle −70° is shown in 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.

Lost Net

(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?).

Caught in a Net, but then it went down ...

It still happens for whatever reason. Too many people taxing the resources at the same time? It's frustrating when it goes out during class when your lesson depends on it.

I've actually said the above line out loud in class. I don't know if my students got the reference or not.

All you can do is laugh, because it better than the other alternative.

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.