Geometry Problems of the Day (Geometry 1 Regents, January 2024 Part I)

This exam was adminstered in January 2024.

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January 2024 Geometry Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


1. Which expression is equal to sin 30°?

(1) tan 30°
(2) sin 60°
(3) cos 60°
(4) cos 30°

Answer: (3) cos 60°

The sin of x degrees is equal to the cos of (90 - x) degrees because what is opposite one angle is adjacent to the complementary angle.

In a 30-60-90 degree right triangle, the sine ratio for the 30 degree angle will use the same leg (and hypotenuse) as the cosine of the 60 degree angle.

2. In the diagram of △ SRA below, KP is drawn such that ∠ SKP ≅ ∠ SRA.

If SK = 10, SP = 8, and PA = 6, what is the length of KR, to the nearest tenth?

(1) 4.8
(2) 7.5
(3) 8.0
(4) 13.3

Answer: (2) 7.5

First of all, this question, as it appears on the NYS Eduction Regents website, IS WRONG. The two triangles are not and cannot be congruent unless they are literally the same triangle. They can however be SIMILAR which would have the symbol ~ instead of "≅". This response assumes similarity.

If the triangles are simmialr then the corresponding sides will be proportional.

SK / SP = KR / PA

10 / 8 = x / 6

8x = 60

x = 7.5, which is choice (2).

Since it is not stated that this is an isosceles trapezoid, the diagonals (not drawn) are not necessarily congruent. Eliminate Choice (2).

In a trapezoid, the bases do not have to be congruent. In fact, in standard HS Geometry, the bases cannot be congruent, because that would make it a parallelogram instead of a trapezoid. (When you get to advanced mathematics, parallelograms will become special cases of trapezoids the way that equilateral triangles are also isosceles.)

3.A rectangle is graphed on the set of axes below.

A reflection over which line would carry the rectangle onto itself?

(1) y = 2
(2) y = 10
(3) y = 1/2 x - 3
(4) y = -1/2 x + 7

Answer: (1) y = 2

This rectangle has two lines of symmetry: the horizontal and veritcal lines that split the rectangle in half.

The horizontal line is y = 2, which is Choice (1) and the vertical line is x = 10, which is not listed.

The other two lines are the equations for the diagonals of the rectangle. However, reflecting a rectangle over its diagonal will not carry the rectangle onto itself unless the rectangle is a square.

4. The surface of the roof of a house is modeled by two congruent rectangles with dimensions 40 feet by 16 feet, as shown below.

Roofing shingles are sold in bundles. Each bundle covers 33 1/3 square feet. What is the minimum number of bundles that must be purchased to completely cover both rectangular sides of the roof?

(1) 20
(2) 2
(3) 39
(4) 4

Answer: (3) 39

How much area do the two rectangle comprise? Divide that by 33 1/3 and round up so that there is enough to finish the job.

The area is 2 * 40 * 16 = 1280.

1280 / 33 1/3 = 38.4, which rounds up to 39.

The answer is Choice (3).

5. Which equation represents a line that is perpendicular to the line whose equation is y - 3x = 4?

(1) y = -1/3 x - 4
(2) y = 1/3 x + 4°
(3) y = -3x + 4
(4) y = 3x - 4

Answer: (1) y = -1/3 x - 4

If they are perpendicular, then the slopes must be inverse reciprocals. Since all four choices have different slopes, that is all we need to know.

The line y - 3x = 4 can be rewritten as y = 3x + 4, which has a slope of 3.

Any line perpendicular to this line must have a slope of -1/3. The only choice is Choice (1).

6. A vertical mine shaft is modeled in the diagram below. At a point on the ground 50 feet from the top of the mine, a ventilation tunnel is dug at an angle of 47°.

What is the length of the tunnel, to the nearest foot?

(1) 47
(2) 54
(3) 68
(4) 73

Answer: (4) 73

Notice that they are looking for the tunnel (the hypotenuse) and not the length of the mine (the opoosite leg).

You need to use the cosine ratio to find the missing hypotenuse.

Cos 47 = 50 / x

So x = 50 / cos 47 = 73.3, which is about 73

7. On the set of axes below, △BLU has vertices with coordinates B(-3,-2), L(-2,5), and U(1,1).

What is the area of △BLU?

(1) 11
(2) 12.5
(3) 14
(4) 17.1

Answer: (2) 12.5

LU has a slope of -4/3 and UB has a slope of 3/4, which means that the two lines are perpendicular and that their lengths can be used as the base and altitude of the triangle.

Also if you use either Distance Formula or Pythagorean Theorem, you will find that the length of LU and UB are both 5. You could also have drawn right triangles and recognized 3-4-5 triangles as the most common right triangle that gets used in problems of this type all the time!

So A = 1/2 (5) (5) = 12.5, which is Choice (2).

8. OIn the diagram below, △CAR is mapped onto △BUS after a sequence of rigid motions.
If AR = 3x + 4, RC = 5x - 10, CA = 2x + 6, and SB = 4x - 4, what is the length of SB?

(1) 6
(2) 16
(3) 20
(4) 28

Answer: (3) 20

If CAR is mapped onto BUS, then SB corresponds to RC and is congruent to RC, which means their lengths are equal. So:

5x - 10 = 4x - 4

x = 6

If x = 6, then SB = 4(6) - 4 = 20, which is Choice (3).

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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 (Algebra 1 Regents, January 2024 Part I)

The following questions appeared on the January 2024 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, January 2024

Part I: Each correct answer will receive 2 credits.


17.If f(x) = 2x + 6 and g(x) = |x| are graphed on the same coordinate computations. plane, for which value of x is f(x) = g(x)?

1) 6
2) 2
3) -2
4) -6

Answer: 3) -2

Set the two functions equal to each other and solver for x.

f(x) = g(x)
2x + 6 = |x|
2x + 6 = -x or 2x + 6 = x
6 = -3x or 6 = -x
-2 = x or -6 = x

Next check to see if the answer is a valid solution.

2(-2) + 6 ?= |-2|
2 = 2

2(-6) + 6 ?= |-6|
-6 =/= 6

Discard x = -6 as a solution. Only x = -2 is a solution, which is Choice (3).

Alternatively, you could've just substituted each choice into both equations until you found a match.

18. What is the solution to the inequality 2x - 7 > 2.5x + 3?

1) x > -5
2) x < -5
3) x > -20
4) x < -20

Answer: 4) x < -20

Use inverse operations to isolate x.

2x - 7 > 2.5x + 3

-.5x > 10

x < -20

Don't forget to flip the inequality symbol with dividing both sides by a negative number.

19. Three expressions are written below.

A. (2xy²)³
B. (2x)³y⁶
C. (2x²y²)(4xy³) Which expressions are equivalent to 8x³y⁶?

1) A and B, only
2) B and C, only
3) A and C, only
4) A, B, and C

Answer: 1) A and B, only

Use the Laws of Exponents and evaluate the three expressions.

(2xy²)³ is 8x³y⁶, which eliminates Choice (2).

(2x)³y⁶ is 8x³y⁶

(2x²y²)(4xy³) is 8x³y⁵, which eliminates Choices (3) and (4).

Choice (1) remains.

20. Joe deposits $4000 into a certificate of deposit (CD) at his local bank. The CD earns 3% interest, compounded annually. The value of the CD in x years can be found using the function

1) f(x) = 4000 + 0.3x
2) f(x) = 4000 + 0.03x
3) f(x) = 4000(1.3)^x
4) f(x) = 4000(1.03)^x

Answer: 4) f(x) = 4000(1.03)^x

Choices (1) and (2) show simple interest, not compounded interest.

Choice (1) and (3) show 30% interest, not 3%, with is 0.03 as a decimal.

The correct answer is Choice (4).

21. When factored completely, -x³ + 10x² + 24x

1) -x(x + 4)(x - 6)
2) -x(x - 4)(x - 6)
3) -x(x + 2)(x - 12)
4) -x(x - 2)(x + 12)

Answer: 3) -x(x + 2)(x - 12)

Factor the -x from each term and then factor the remaining quadratic expression.

-x³ + 10x² + 24x

-x(x² - 10x² - 24)

-x(x - 12)(x + 2)

Notice that when factoring out the -x, the signs of the coefficients of each of the terms were changed.

The correct answer is Choice (3).

22. When the temperature is 59°F, the speed of sound at sea level is 1225 kilometers per hour. Which process could be used to convert this speed into feet per second?

Answer: 2) [See image]

Kilometers needs to be converted to feet, so feet needs to be in the top of a fraction and kilometers in the bottom to cancel out with the given unit.

Hours needs to be converted to seconds, so seconds needs to be in the bottom of a fraction and hours needs to be in the top to cancel out with the given unit.

Doing unit analysis on the choices:

Choice (1) has miles twice on top and both feet and seconds on the bottom. Eliminate Choice (1).

Choice (2) cancels all the units except for feet on top and seconds on the bottom. This is the correct answer.

Choice (3) has kilometers twice on top and miles twice on the bottom, so they won't cancel out. Eliminate Choice (3).

Choice (4) has minutes twice on top and hours twice on the bottom, so they won't cancel. Eliminate Choice (4).

23. The zeros of a polynomial function are -2, 4, and 0. What are all the factors of this function?

1) (x + 2) and (x - 4)
2) (x - 2) and (x + 4)
3) x, (x + 2) and (x - 4)
4) x, (x - 2) and (x + 4)

Answer: 3) x, (x + 2) and (x - 4)

There are three zeroes, so there must be three factors. Remember that you flip the signs when writing the factors. If x = 4, it is because x - 4 = 0.

So the correct answer is x, (x + 2) and (x - 4), which is Choice (3).

24. What is the range of the function f(x) = (x - 4)² + 1?

1) x > 4
2) x > 4
3) f(x) > -1
4) f(x) > -1

Answer: 4) f(x) > -1

The range of the function refers to all of the values that the output of the function could possibly have. That is, the y values or the f(x) values. This means that you can eliminate Choices (1) and (2).

The given function is quadratic. It is a parabola with a minimum point (its vertex) at the point (4,-1). If you didn't realize this by looking at the funtion, you could put it in a graphing calculator and see it.

Since (4,-1) is a point on the graph, Choice (3) must be eliminated. The range is all values greater than or equal to -1.

This is Choice (4).

End of Part I.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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 (Algebra 1 Regents, January 2024 Part I)

The following questions appeared on the January 2024 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, January 2024

Part I: Each correct answer will receive 2 credits.


9. Mrs. Smith’s math class surveyed students to determine their favorite flavors of soft ice cream. The results are shown in the table below.

	Chocolate	Vanilla	Twist
Juniors	42	27	45
Juniors	67	42	21

Of the students who preferred chocolate, approximately what percentage were seniors?

1) 27.5
2) 44.7
3) 51.5
4) 61.5

Answer: 4) 61.5

There were 42 + 67 = 109 who preferred Chocolate. Of those, 67/109 were seniors. Even without a calculator, you can see that there is only one possible answer, which is 61.5.

You don't need to know the number of juniors and seniors in the survey. You only need to know how many liked chocolate.

10. If f(x) = x² + 2x + 1 and g(x) = 3x + 5, then what is the value of f(1) - g(3)?

1) 10
2) 8
3) -10
4) -8

Answer: 3) -10

Substitute into each function and evaluate.

f(1) - g(3) = (1)² + 2(1) + 1 - (3(3) + 5) = 1 + 2 + 1 - 9 - 5 = -10, which is Choice (3).

11. Which function has the largest y-intercept?

Answer: 3) g(x) = |x| + 3

The y-intercept is the value of y when x = 0.

In Choice (1), the y-intercept is -1.

In Choice (2), the y-intercept is 2.

In Choice (3), the y-intercept is 3.

In Choice (4), the y-intercept is 1.

The largest of these is 3, which is Choice (3).

12. Two texting plans are advertised. Plan A has a monthly fee of $15 with a charge of $0.08 per text. Plan B has a monthly fee of $3 with a charge of $0.12 per text. If t represents the number of text messages in a month, which inequality should be used to show that the cost of Plan A is less than the cost of Plan B?

1) 15 + 0.08t < 3 + 0.12t
2) 15 + 0.08t > 3 + 0.12t
3) 15t + 0.08 < 3t + 0.12
4) 15t + 0.08 > 3t + 0.12

Answer: 1) 15 + 0.08t < 3 + 0.12t

The monthly fee is charged once and the per text charge gets the variable. Eliminate Choices (3) and (4).

Plan A < Plan B is Choice (1).

13. The function f(x) is graphed on the set of axes below.

What is the equation of the axis of symmetry for f(x)?

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

Answer: 1) x = -1

The axis of symmetry is a vertical line that goes through the vertex and cuts the parabola in half so that the left side is a mirror image of the right side.

The equation for any vertical line starts with "x =", so eliminate Choices (3) and (4).

The vertex is at (-1,-3), so the axis of symmetry is x = -1, which is the x-coordinate of the vertex. This is Choice (1).

14. What is the degree of the polynomial 5x - 3x² - 1 + 7x³?

1) 1
2) 2
3) 3
4) 5

Answer: 3) 3

The degree of a polynomial is the largest exponent. (Note: it is not the exponent of the first term unless the polynomial is written in standard form.)

The largest exponent is 3, which is Choice (3).

15. The product of (x² + 3x + 9) and (x - 3) is

1) x³ -27
2) x² + 4x + 6
3) x³ - 6x² - 18x - 27
4) -6x⁴ + x³ - 18x² - 27

Answer:1) x³ -27

You can use the Distributive Property or the Area Model/Box Method to find the solution, but you don't need to.

If you look at the first terms of the two expressions, you have (x²)(x) which is x³, which will be the leading term of the product.

This means you can eliminate Choices (2) and (4).

Using the box method gets you this:

When the graph is shifted to the left, that means you are adding a value to the inside of the parentheses which gets squared.

	x2	3x	9
x	x3	3x2	9x
-3	-3x2	-9x	-27

Adding like terms gets you x³ + 0 x² + 0x - 27, which is Choice (1).

16. The solution to (2/3)(3 - 2x) = 3/4 is

1) -11/8
2) 5/8
3) -33/16
4) 15/16

Answer: 4) 15/16

Use inverse operations to isolate the varaible.

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

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

-2x = 9/8 - 3

x = (9/8 - 24/8) / (-2)

x = (-15/8) / (-2) = (-15/-16) = 15/16

This is Choice (4).

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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 (Algebra 1 Regents, January 2024 Part I)

The following questions appeared on the January 2024 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, January 2024

Part I: Each correct answer will receive 2 credits.


1. The graph below represents a dog walker’s speed during his 30-minute walk around the neighborhood.

Which statement best describes what the dog walker was doing during the 12-18 minute interval of his walk?

1) He was walking at a constant rate.
2) He was increasing his speed.
3) He was decreasing his speed.
4) He was standing still.

Answer: 1) He was walking at a constant rate.

He is walking at a constant rate of 1.5 miles per hour. Were he standing still, his speed would be zero. Increasing his speed causing the graph to have a positive slope, and decreasing his speed will give it a negative slope.

2. Given the relation: {(0,4), (2,6), (4,8), (x,7)} Which value of x will make this relation a function?

1) 0
2) 2
3) 6
4) 4

Answer: 3) 6

It can only be a function if none of the x values repeat. The domain already has 0, 2, and 4 in it, so they cannot be repeated. The only possible choice given is 6.

3. The Speedy Jet Ski Rental Company charges an insurance fee and an hourly rental rate. The total cost is modeled by the function R(x) = 30 + 40x. Based on this model, which statements are true?

I. R(x) represents the total cost.
II. x is the number of hours rented.
III. $40 is the insurance fee.
IV. $30 is the hourly rental rate.

1) I, only
2) I and II, only
3) I, III, and IV, only
4) I, II, III, and IV

Answer: 2) I and II, only

The total cost is modeled by the function R(x), so statement I is true.

The cost has an hourly rate and the number of hours is the variable, x. So statement II is also true.

The insurance fee is charge a single time and does not get multiplied by the number of hours. So the insurance fee must be $30, not $40. Statement III is false.

The hourly rental rate gets the variable for the number of hours, so it must be $40 and not $30. Statement IV is false.

4. The eleventh term of the sequence 3, -6, 12, -24, …, is

1) -3072
2) -6144
3) 3072
4) 6144

Answer: 3) 3072

You can see that the numbers are doubling except that the sign is also changing. This is a geometric sequence with a common ratio of -2. You can see that the odd terms are positive and the even terms are negative, so Choices (1) and (2) can be eliminated.

You could put -24 into your calculator and multiply by -2 to find the fifth term and then multiply the answer six more times.

48, -96, 192, -384, 768, -1536, 3072

Choice (3) is the correct answer.

You could also use the formula to find the nth term of a geometric sequence: a_n = a₁r^{n - 1}.

a₁₁ = (3)(2)^(11-1) = 3072.

5. Which situation represents exponential growth?

1) Aidan adds $10 to a jar each week.
2) A pine tree grows 1.5 feet per year.
3) Ella earns $20 per hour babysitting.
4) The number of people majoring in computer science doubles every 5 years.

Answer: 4) The number of people majoring in computer science doubles every 5 years.

Exponential growth means that it has a common ratio. There is a some number that is multiplied from term to term. It is not growing by a constant amount.

Adding $10 per week is constant. It's a linear equation. Eliminate Choice (1).

Growing 1.5 feet per year is constant. It's also linear. Eliminate Choice (2).

Earning $20 per hour is a constant amount. Linear. Eliminate Choice (3).

Doubling every 5 years has a constant rate of growth but the amount changes each time. Choice (4) is the answer.

6. The expression (-x² + 3x - 7) - (4x² + 5x - 2) is equivalent to

1) -5x² - 2x - 9
2) -5x² - 2x - 5
3) -5x² + 8x - 9
4) -5x² + 8x - 5

Answer: 2) -5x² - 2x - 5

Combine the like terms. Make sure to distribute the subtraction to all the terms in the second polynomial.

(-x² + 3x - 7) - (4x² + 5x - 2)
-x² + 3x - 7 - 4x² - 5x + 2

-5x² - 2x - 5

This is Choice (2).

7. If f(x) = x², which function is the result of shifting f(x) 3 units left and 2 units down?

1) g(x) = (x + 2)² - 3
2) h(x) = (x - 2)² + 3
3) j(x) = (x + 3)² - 2
4) k(x) = (x - 3)² + 2

Answer: 3) j(x) = (x + 3)² - 2

If the graph is being shifted down 2 units, there there will be a -2 at the end. There is only one choice, and that's Choice (3).

When the graph is shifted to the left, that means you are adding a value to the inside of the parentheses which gets squared.

8. An equation used to find the velocity of an object is given as v² = u² + 2as,where u is the initial velocity, v is the final velocity, a is the acceleration of the object, and s is the distance traveled. When this equation is solved for a, the result is

1) a = (v² u²) / (2s)
2) a = (v² - u²) / (2s)
3) a = v² - u² - 2s
4) a = 2s(v² - u²)

Answer: 4) the number of time periods

Use inverse operations to isolate the varaible, which is a in this case.

v² = u² + 2as
v² - u² = 2as
(v² - u²) / (2s) = a

This is Choice (2).

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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.

Total Eclipse

(Click on the comic if you can't see the full image.)

(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Happy Eclipse Day

Back in 2017, I told myself that I was going to get to see the next one in 2024 because it was going to happen so close to home. Well, that didn't happen, and I pretty much knew that it wouldn't. Maybe had it been in the summer, I might've taken the trip, but it was during the school year. It's debatable if I could've arranged it. It's not a once-in-a-lifetime event, but it could be the last-in-my-lifetime. Lots of people are pushing back asking what the big deal is, but seeing the entire sun blocked out isn't anything like just a part of it. They aren't science geeks. Then again, neither am I, but I can appreciate them.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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.

New Book: COVER REVEAL Burke's Lore Briefs #2 Portrait of a Lady Vampire & Other Vampiric Cravings

A new book in the Burke's Lore Briefs series is coming next week. In the meantime, here is the cover.

What do you think?

(x, why?) Mini: Septagon

(Click on the comic if you can't see the full image.)

(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Every slice sends you closer to prism!

It's a multi-layered story.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story. 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.

January 2024 Algebra 2 Regents Part IV

This exam was adminstered in January 2024.

More Regents problems.

Algebra 2 January 2024

Part IV: A correct answer will receive 6 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.


37.A manufacturer of sweatshirts finds that profits and costs fluctuate depending on the number of products created. Creating more products doesn’t always increase profits because it requires additional costs, such as building a larger facility or hiring more workers. The manufacturer determines the profit, p(x), in thousands of dollars, as a function of the number of sweatshirts sold, x, in thousands. This function, p, is given below.

p(x) = -x³ + 11x² - 7x - 69

Graph y = p(x), over the interval 0 < x < 9, on the set of axes below

Over the given interval, state the coordinates of the maximum of p and round all values to the nearest integer. Explain what this point represents in terms of the number of sweatshirts sold and profit.

Determine how many sweatshirts, to the nearest whole sweatshirt, the manufacturer would need to produce in order to first make a positive profit. Justify your answer.

Answer:

Put the function in your graphing calculator and get the Table of Values. Plot the points on the graph from 0 through 9. Connect them but DO NOT add any arrows that go beyond the specified range. Notice that each box on the y-axis is 15 units, while each box on the x-axis is one.

The points you need to plot are (0,-69), (1,-66), (2,-47), (3,-18), (4,15), (5,46), (6,69), (7,78), (8,67), (9,30).

Your graph will look like this:

The coordinates maximum value, to the nearest interger, are (7,78). This means that for 7,000 sweatshirts the profit will be $78,000.

Using the calculator, you can find the zero of the function to be (3.5488,0). Since the function represents thousands of sweatshirts, then this is 3548.8 sweatshirts. That means that the first positive profit will be at 3549 sweatshirts.

End of Exam

How did you do?

More to come. Comments and questions welcome.

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I also write Fiction! You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story. 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.

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(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

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I also write Fiction! You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story. 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.