Algebra 2 Problems of the Day (Algebra 2 Regents, August 2024 Part I)

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Algebra 2 Regents

Part I

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


17. Given q(x) = 2log(x) and r(x) = (x - 2)³ - 4, what is a solution of q(x) = r(x) to the nearest tenth?

(1) 1.1
(2) 3.7
(3) 3.9
(4) 4.3

Answer: (2) 3.7

This is a MULTIPLE-CHOICE question. Do not attempt to solve it algebraically. It will NOT be a fun time.

You can graph both equations on your calculator and look for the point where they intersect. (There's actually a function on the calculator that will help you find it if you know how to use it.)

You can also find the values of q(1.1), r(1.1), q(3.7), r(3.7), etc to find the ones that are the same.

Let's do this just for giggles:

q(1.1) = 0.08, r(1.1) = -4.7

q(3.7) = 1.1, r(3.7) = .91

q(3.9) = 1.2, r(3.9) = 2.9

q(4.3) = 1.7, r(4.3) = 8.2

You will notice that when x = 3.7, the values of q(x) and r(x) are the closest. They differ because of rounding. (I did a little experimenting so that you wouldn't have to, and found that x = 3.725 is pretty close but still not exact.)

Choice (2) is the correct answer.

How would I solve this algebraically? I wouldn't. Simple as that.

If you would like to, please place it in the comments.

18. The volume of a cardboard box can be modeled by V(x), which is the product of the length, width, and height, x. If the length can be represented by L(x) = 18 - 2x and the width can be represented by W(x) = 18 - 2x, then which function represents V(x)?

(1) V(x) = 4x² - 72x + 324
(2) V(x) = 4x³ - 72x² + 324x
(3) V(x) = -3x + 36
(4) V(x) = 4x³ + 324x

Answer: (2) V(x) = 4x³ - 72x² + 324x

It should be "obvious" that since we have to multiply three terms with the variable x in them that the final formula will contain x³ in it.

V = (18 - 2x)(18 - 2x)(x)
= (324 - 36x - 36x + 4x²)(x)
= 324x - 72x² + 4x³

Choice (2) is the correct choice.

Choice (4) is missing the middle term. Choice (1) is the Area of the base. Choice (3) is nonsense because it just added the three expressions.

19. The expression 8^x/2 * 8^x/3 is equivalent to

(1) ⁶√(8^5x)
(2) 64^(5x/6)
(3) ⁵√(8^2x)
(4) 64^(x2/6)

Answer: (1) ⁶√(8^5x)

Keep the base and add the exponents x/2 + x/3.

The common denominator is 6, so 3x/6 + 2x/6 = 5x/6, and the expression will be 8^5x/6, which is not one of the listed choices.

However, the one-sixth power is the same as taking the sixth root of the expressions.

Therefore, 8^5x/6 is the same as (8^5x)^(1/6 which is the same as ⁶√(8^5x). This is Choice (1).

20. If θ is an angle in standard position whose terminal side passes through the point (-3,-4), which statement is true?

(1) sec θ > 0 and tan θ > 0
(2) sec θ < 0 and tan θ < 0
(3) sec θ > 0 and tan θ < 0
(4) sec θ < 0 and tan θ > 0

Answer: (4) sec θ < 0 and tan θ > 0

Point (-3,-4) is in Quadrant III where sin is negative, cos is negative, and tan is positive. Eliminate Choices (2) and (3).

Sec = 1/cos and has the same sign and cos, which is negative in Quadrant III. So the correct answer is Choice (4).

21. What is the value of y for the system shown below? 3x + 4y - 5z = -27
2x + 3y - z = -3
6x - y + 4z = 3

(1) -27
(2) 6
(3) 3
(4) -3

Answer: (3) 3

Solve the system of equations by eliminating the x term first since 2 and 3 are factors of 6.

3x + 4y - 5z = -27
2x + 3y - z = -3
6x - y + 4z = 3

6x + 8y - 10z = -54
6x + 9y - 3z = -9
6x - y + 4z = 3

First 2 equations: y + 7z = 45
Last 2 equations: 10y - 7y = -12
Combined: 11y = 33
y = 3

Choice (3) is the correct answer.

22. The number of employees who work nights and weekends at a department store is summarized in the table below.

Let N represent the event “works nights” and let W represent the event “works weekends.” Based on the table, are N and W independent events?

(1) Yes, because P(N) • P(W) = P(N ∩ W).
(2) Yes, because P(N) • P(W) =/= P(N ∩ W).
(3) No, because P(N) • P(W) = P(N ∩ W).
(4) No, because P(N) • P(W) =/= P(N ∩ W).

Answer: (1) Yes, because P(N) • P(W) = P(N ∩ W).

The answer is either Yes because they are equal or No because they are not equal. Multiply the probabilities.

There are a total of 120 employees. P(N) = 20/120 and P(W) = 48/120, and (20/120)(48/120) = (1/6)(2/5) = 1/15.

P(N ∩ W) = 8/120 = 1/15.

So P(N) • P(W) = P(N ∩ W).

The correct answer is Choice (1).

23. Which expression is equivalent to x⁸ - y⁸?

(1) (x - y)⁸
(2) (x² + y²)²(x² - y²)²
(3) (x⁴ + y⁴)(x² + y²)(x + y)(x - y)
(4) (x + y)⁴(x - y)⁴

Answer: (3) (x⁴ + y⁴)(x² + y²)(x + y)(x - y)

The given expression is a Difference of Perfect Squares and canb be factored into two conjugates. One of those factors will still be a Difference of Perfect Squares and can be factored again. And then again.

x⁸ - y⁸
(x⁴ + y⁴)(x⁴ - y⁴)
(x⁴ + y⁴)(x² + y²)(x² - y²)
(x⁴ + y⁴)(x² + y²)(x + y)(x - y)

The correct answer is Choice (3).

If you factor Choice (1), you will find that it is equivalent to Choice (4), so both could be eliminated.

24. A research assistant receives a first year salary of $90,000 and a 2% annual raise throughout the first ten years of employment. In total, how much money will be earned over the first ten years, to the nearest dollar?

(1) $91,837
(2) $109,709
(3) $877,917
(4) $985,475

Answer: (4) $985,475

Using the formula for a Geometric Series: S_n = (a₁ - a₁rⁿ) / (1 - r), we can calculate the answer.

S₁₀ = (90000 - 90000*1.02¹⁰) / (1 - 1.02) = 985474.89, which is Choice (4).

End of Part I. How did you do?

Comments and questions welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra 2 Problems of the Day (Algebra 2 Regents, August 2024 Part I)

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Algebra 2 Regents

Part I

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


9. The asymptote of the graph of f(x) = 5 log(x + 4) is

(1) y = 6
(2) x = -4
(3) x = 4
(4) y = 5

Answer: (2) x = -4

First of all, you can graph this and look. Second, the asymptote will be a vertical line, which means it starts with "x = ", so eliminate Choices (1) and (4).

The asymptote of f(x) = log x is the y-axis, which is x = 0. Multiplying by 5 will not change the asymptote. However, adding 4 inside the parentheses, that is log(x + 4), will move the asymptote 4 units to the left, to x = -4. Choice (2) is the correct answer.

10. The probability of having math homework is 1/3 and the probability of having English homework is 1/7. The probability of having math homework or having English homework is 9/21. What is the probability of having math homework and having English homework?

(1) 19/21
(2) 1/5
(3) 1/21
(4) 10/21

Answer: (3) 1/21

P(M) + P(E) - P(M|E) = 1/3 + 1/7 - 9/21 = 7/21 + 3/21 - 9/21 = 1/21

Choice (3) is the correct choice.

11. The solution set to the equation x - 1 = √(2x + 6) is

(1) {5, -1}
(2) {5}
(3) {-1}
(4) { }

Answer:(2) {5}

You can check each of the listed answers to see if either, or both work.

For x = 5, 5 - 1 = 4 and √(2(5) + 6) = 4, so the answer is either Choice (1) or (2).

For x = -1, -1 -1 = -2, which cannot be the result of a square root. Eliminate Choices (1) and (3).

Choice (1) is the correct answer.

If you tried to solve it algebraically:

x - 1 = √(2x + 6)
x² - 2x + 1 = 2x + 6
x² - 4x - 5 = 0
(x - 5)(x + 1) = 0
x = 5 or x = -1

However, when we test the answers, we see that x = -1 must be discarded as extraneous, and only x = 5 is correct.

12. Given x > 0, the expression (1 / x^-2)^(-3/4) is equivalent to

(1) x √(x)
(2) 1 / (x √(x))
(3) ∛x²
(4) 1 / (∛x²)

Answer: (2) 1 / (x √(x))

Use the rules for exponents to evaluate the expression.

(1 / x^-2)^(-3/4)
(x²)^(-3/4)
(x^(-3/2))
1 / x^3/2
1 / √(x³ = 1 / x √(x)

Choice (2) is the correct answer.

13. The graph of which function has a period of 3?

(1) y = -7sin(2π/3 x) - 5
(2) y = -7sin(3π/2 x) + 9
(3) y = -7sin(3x) - 5
(4) y = 3sin(π x) + 9

Answer: (1) y = -7sin(2π/3 x) - 5

Once again, you can graph these equations and look to see which is correct.

The period of y = A sin(x) + C is 2π. The period of y = A sin(Bx) + C is 2π/B.

In Choice (1), 2π/ (2π/3) = 3. This is the correct answer.

In Choice (2), 2π/ (3π/2) = 4/3. Eliminate Choice (2).

In Choice (3), 2π/ 3 is not equal to 3. Eliminate Choice (3).

In Choice (4), 2π/ (π) = 2. Eliminate Choice (4).

14. Which graph could represent a 4th degree polynomial function with a positive leading coefficient, 2 real zeros, and 2 imaginary zeros?

Answer: (1) [See image]

A positive leading coefficient on a fourth-degree polynomial means that the end behavior is toward positive infinity on both ends. Eliminate Choices (3) and (4).

Two real zeroes means that the graph crosses the x-axis in two locations, which eliminates Choice (4) and leaves Choice (1).

Since there are only two real zeroes, there must be two imaginary zeroes as well.

15. Given i is the imaginary unit, which expression is equivalent to 5i(2x + 3i) - x√(-9)?

(1) 15 + 13xi
(2) -15 + 13xi
(3) 15 + 7xi
(4) -15 + 7xi

Answer: (4) -15 + 7xi

Use the Distributive Property and the Order of Operations. Don't forget that i² = -1 and that √(-1) = i.

5i(2x + 3i) - x√(-9)
10xi + 15i² - 3xi
-15 + 7xi

The correct answer is Choice (4).

16. 6 What is the focus of the parabola 8(y + 2) = (x + 5)²?

(1) (-5,0)
(2) (-5,-4)
(3) (5,0)
(4) (5,4)

Answer: (1) (-5,0)

The vertex form of a parabola is y = a(x - h) + k, and the focus of the parabola is at point (h, k + 1/(4a)). Rewrite the equation in vertex form.

8(y + 2) = (x + 5)²

y + 2 = (1/8)(x + 5)²

y = (1/8)(x + 5)² - 2

If a = 1/8, then 1/(4a) = 1/(4(1/8)) = 1/(1/2) = 2.

So (h, k + 1/(4a)) = (-5, -2+2) = (-5,0).

More to come. Comments and questions welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra 2 Problems of the Day (Algebra 2 Regents, August 2024 Part I)

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Algebra 2 Regents

Part I

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


1. A grocery store owner wonders how many customers bring reusable bags to the store. An employee stands at the store entrance for two hours and counts the number of people bringing in reusable bags. This type of study is best classified as

(1) a census
(2) an experiment
(3) an observational study
(4) a survey

Answer: (3) an observational study

This is a case of an observational study. I hope it's self-explanatory.

No questions are being asked, so it isn't a census or a survey. No experiment is being conducted.

Choice (3) is the correct choice.

2. The graph of y = 2^x - 4 is positive on which interval?

(1) (-∞,∞)
(2) (2,∞)
(3) (-0,∞)
(4) (-4,∞)

Answer: (2) (2,∞)

The graph will cross the x-axis (y=0) when 2^x = 4, which occurs when x = 2.

When x > 2, the graph will be above the x-axis.

You can also put this equation into your graphing calculator and look at the screen and the table of values.

The correct answer is Choice (2).

3. Tim deposits $300 into a savings account. The annual interest rate is 2.7% and compounds monthly. He uses the equation A = 300(1 + 0.027/12)^12t to determine how much money, A, he will have after t years. Which equation is equivalent to Tim’s equation?

(1) A = 300[(1.00225)¹²]^t
(2) A = 300[(0.08558)¹²]^t
(3) A = 300[1 + (0.027/12)^12t)]
(4) A = (300)^12t(1)^12t + (0.027/12)^12t

Answer: (1) A = 300[(1.00225)¹²]^t

Divide 0.027 by 12 and you'll get 0.00225. Add 1 and you have 1.00225. Choice (1) is the correct answer.

The other answers are a bit silly and violate a bunch of rules of exponents.

Choice (2) splits the 12 and t exponents like Choice (1) does, but the growth rate becomes a decay rate. I have no idea where .08558 came from.

Choice (3) applied the exponent 12t only to the fraction. This is like saying that (1 + 2)² is equal to (1 + 2²), which is silly.

In Choice (4), the 300 gets the exponent of 12t, which is crazy even without all the other problems. This should only be selected by students not even reading the question.

4. Which equation is true for all real values of x?

(1) x⁴ + x = (x + 1)(x³ - x² + x)
(2) x⁴ + x = (x + 1)(x³ + x)
(3) x⁴ + x = (x² + x)²
(4) x⁴ + x = (x + 1)(x³ + x² + x)

Answer: (1) x⁴ + x = (x + 1)(x³ - x² + x)

You can multiply the polynomials on the right side to see which one gives you x⁴ + x. Or you can put each of these into the graphing calculator to see which ones give you the same graph.

y = x⁴ + x
y = (x + 1)(x³ - x² + x)
y = (x + 1)(x³ + x)
y = (x² + x)²
y = (x + 1)(x³ + x² + x)

Choice (1) has the same graph and table of values.

Algebraically:

(x + 1)(x³ - x² + x)
(x)(x³ - x² + x) + (x³ - x² + x)
x⁴ - x³ + x² + x³ - x² + x
x⁴ + x

5. The solution of

x/(x + 3) + 2/(x - 4) = (2x + 27)/(x² - x - 12) is

(1) -3
(2) -7
(3) 3
(4) 7

Answer: (4) 7

You can work backward from the answers to see which one works. You can immediately see that x = -3 is not a real answer.

You can graph y = x/(x + 3) + 2/(x - 4) - (2x + 27)/(x² - x - 12) and look for a value of y = 0.

Or you can solve this algebraically once you notice that (x + 3)*(x - 4) = x² - x - 12.

x/(x + 3) + 2/(x - 4) = (2x + 27)/(x² - x - 12)

(x - 4)/(x - 4) * x/(x + 3) + (x + 3)/(x + 3) * 2/(x - 4) = (2x + 27)/(x² - x - 12)

(x - 4)(x)/(x² - x - 12) + (x + 3)(2)/(x² - x - 12) = (2x + 27)/(x² - x - 12)

(x - 4)(x) + (x + 3)(2) = 2x + 27

x² - 4x + 2x + 6 = 2x + 27
x² - 4x - 21 = 0
(x - 7)(x + 3) = 0
x = 7 or x = -3

We've already discarded x = -3 as an answer, so x = 7 is the only solution, and that is Choice (4).

If you had graphed the above equation, you would have gotten y = 0 for x = 7.

6. The cost, in dollars, of a single-ride fare in the New York City subway in the years since 1904 is listed in the table below.

Which equation best models the cost of a single-ride fare based on these data?

(1) y = 0.0375(1.0392)^x
(2) y = 1.0392(0.0375)^x
(3) y = 0.0234x - .0487
(4) y = -0.179 + 0.356 ln(x)

Answer: (1) y = 0.0375(1.0392)^x

You can see that the fare went up only 10 cents in the first fifty years but then increased 35 cents in the next twenty or so and $1.00 more in the next twenty. Clearly this is an exponential function and not a linear function.

Only Choices (1) and (2) are exponential functions. Choice (2) shows decay, not growth, which leaves Choice (1) as the correct answer.

7. Which expression is equivalent to (6x⁴ + 4x³ + x + 200) / (x + 2)

(1) 6x² - 8x + 17 + 166/(x + 2)
(2) 6x² - 16x + 33 + 266/(x + 2)
(3) 6x³ + 16x² + 32x + 65 + 330/(x + 2)
(4) 6x³ - 8x² + 16x - 31 + 262/(x + 2)

Answer: (4) 6x³ - 8x² + 16x - 31 + 262/(x + 2)

A fourth-order polynomial divided by (x + 2) becomes a third-order polynomial. Eliminate Choices (1) and (2).

Once again, you can put the original equation along with Choices (3) and (4) into your graphing calculator to see which has the same table of values.

Or you can choose a random value for x, such as 8 or 10, and see which expression gives the same value.

Or you can divide:

Once you get this far, you only have one option left, and that is Choice (4).

If you want to continue, just to make sure, the rest of the division looks like this:

There is a remainder of 262 which means that the final term will be 262/(x + 2).

8. The solution to the equation 6(2^{x + 4}) = 36 is

(1) -1
(2) ln 36 / ln 12 - 4
(3) ln(3) - 4
(4) ln 6 / ln 2 - 4

Answer: (4) ln 6 / ln 2 - 4

A quick check can eliminate x = -1 because 2³ = 8 and 6(8) is not equal to 36.

Use inverse operations:

6(2^{x + 4}) = 36
2^{x + 4} = 6
ln 2^{x + 4} = ln 6
(x + 4) ln 2 = ln 6
x + 4 = ln 6 / ln 2
x = ln 6 / ln 2 - 4

This is Choice (4).

More to come. Comments and questions welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

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

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Geometry Regents

Part I

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


17. The line represented by the equation y = 4x + 15 is dilated by a scale factor of 2 centered at the origin. Which equation represents its image?

(1) y = 4x + 15
(2) y = 4x + 30
(3) y = 8x + 15
(4) y = 8x + 30

Answer: (2) y = 4x + 30

The slope of the line will not change because in a dilation the orientation of the image is the same as the pre-image. However, the y-intercept will be twice as far from the origin as the origin line.

Since the original line has a point at (0,15), the image will go through the point (0,30).

Choice (2) is the correct answer.

18. Line segment RH has endpoints R(-4,4) and H(2,-4). Which equation represents a line perpendicular to RH that passes through the point (3,-1)?

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

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

The only difference in the choices is the slope of the line. So you need to find the slope of the original line and then find the slope of the line that is perpendicular to it.

The slope of line RH is △y/△x = the change in y / the change in x = (-4 - 4) / (2 - (-4)) = -8 / 6 = -4/3.

The slope of a line perpendicular to RH would have a slope that was the inverse reciprocal of the slope of RH, which would be +3/4. This is Choice (1).

Check: (-4/3)(+3/4) = (-12/12) = -1. The lines are perpendicualr.

19. In right triangle SNO below, altitude NW is drawn to hypotenuse SO.

Which statement is not always true?

(1) SO/SN = SN/SW
(2) SW/NS = NS/OW
(3) SO/ON = ON/OW
(4) OW/NW = NW/SW

Answer: (2) SW/NS = NS/OW

There are 3 right triangles in the image: the large one and the two smaller one. All three triangles are similar, meaning that they have the same shape, congruent corresponding angles, and corresponding sides that are proportional. Each triangle has a hypotneuse across from the right angle, and a longer leg and a shorter leg across from the larger and smaller acute angles, respectively.

The proportion in Choice (1) compares the hypotenuse / short leg of SON with the hypotenuse / short leg of NSW. Eliminate Choice (1).

The proportion in Choice (2) compares the short leg / hypotenuse NSW with two legs of two different triangles. There is no reason for these to ratios to be equal to each other. Choice (2) is the correct answer.

The proportion in Choice (3) compares the hypotenuse / long leg of SON with the hypotenuse / long leg of NOW. Elimnate Choice (3).

The proportion in Choice (4) compares the long leg / short leg of NOW with the long leg / short leg of NSW. Eliminate Choice (4).

20. A rectangle has a width of 3 and a length of 4. The rectangle is dilated by a scale factor of 1.8. What is the area of its image, to the nearest tenth?

(1) 3.7
(2) 6.7
(3) 21.6
(4) 38.9

Answer: (4) 38.9

The area of the image is Length X scale factor times Width X scale factor.

A = 3 * 1.8 * 4 * 1.8 = 38.88 = 38.9

The correct answer is Choice (4).

Notice that the original rectangle had an area = 3 * 4 = 12, and the image was bigger, so Choices (1) and (2) are just silly. Choice (3) is the product of 3 * 4 * 1.8.

21. In the diagram below of circle P, diameter MD and chord AL intersect at Q, m∠AQD = 46°, and mLD = 124°.

What is m AD?

(1) 36°
(2) 46°
(3) 51°
(4) 92°

Answer: (1) 36°

You have to work your way around the circle to fill in other things you know. For example, since MD is a diameter, then we know that the sum of arcs ML and LD is 180 degrees. Since LD = 124, then ML = 180 - 124 = 56 degrees.

The sum of arc ML and AD is equal to twice the size of angle AQD, so 56 + AD = 92, and AD = 92 - 56 = 36 degrees, which is Choice (1).

Logically, since Q is not the center of the circle, arc AD was going to be less than 46 degrees, and there was only one choice that made since.

The rule that was used above says, basically, that if two intersecting chords intercept two arcs, the vertical angles were the chords intersect will be equal to the average of the two arc sizes. In this case, the arcs are 56 and 36 degrees, which average to 46 degrees.

22. The right prism with a triangular base shown below is cut by a plane perpendicular to its bases.

The two-dimensional shape of the cross section is always a

(1) triangle
(2) rhombus
(3) pentagon
(4) rectangle

Answer: (4) rectangle

Perpendicular to the bases means you're cutting up and down like you're slicing into a wedge of cheese. When you cut it that way, no matter how you slice it, the cross section will be a rectangle composed of a line from the top and bottom planes (parallel to each other) and two vertical lines down two of the three sides of the triangle. Those two vertical lines are also parallel. Two pairs of parallel sides make a parallelogram, but the sides are horizontal and vertical, making right angles, so it's a rectangle.

The correct answer is Choice (4).

If the question had said parallel to the bases, slicing horizontally between the top and bottom bases, then the cross section would have been a triangle.

The other two choices, pentagon and rhombus, are just silly. Why not an octagon?

23. A rectangular fish tank measures 24 inches long, 12 inches wide, and 16 inches high, as modeled in the diagram below.

If the empty tank weighs 25 pounds and the fish tank is filled with water to a height of 14 inches, what is the approximate weight of the tank and water?
[27.7 in³ = 1 pound of water]

(1) 146
(2) 166
(3) 171
(4) 191

Answer: (3) 171

First of all, the height of the tank doesn't matter because there's only 14 inches of water in the tank, so that's the "height" that we need. Second, you have to find the Volume in cubic inches and then convert that to pounds of water.

Volume = L X W X H = (24)(12)(14) = 4032 in³

Weight of the water = (4032 in³) / (27.7 in³ / pound) = 145.56 pounds

Weight of the tank = 25 pounds

Total weight = 146 + 25 = 171, which is Choice (3).

Notice that 146 was one of the incorrect Choices.

24. A circle has a radius of 4.5. What is the measure of the central angle that intercepts an arc whose length is 6.2, to the nearest degree?

(1) 35°
(2) 42°
(3) 64°
(4) 79°

Answer: (4) 79°

Find the circumference of the circle and then find the part of the circle that the arc represents.

C = 2 (3.141592)(4.5) = 28.274328

Divide 6.2/28.274328 = 0.219

Multiply 0.219 * 360° = 78.84° or 79°, which is Choice (4).

End of Part I.

How did you do?

Comments and questions welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

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

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Geometry Regents

Part I

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


9. In right triangle ABC below, m∠C = 90°, AC = 12, and m∠A = 25°.

Which equation is correct for △ABC?

(1) a = 12 / tan 25°
(2) a = 12 tan 25°
(3) c = 12 / tan 25°
(4) c = 12 tan 25°

Answer: (2) a = 12 tan 25°

Side c is the hypotenuse and opposite from angle C. The hypotenuse is not used for the tangent function, so eliminate Choices (3) and (4) immediately.

Tan A = opp/adj = a / 12

Therefore, a = 12 tan A = 12 tan 25°, which is Choice (2).

10. Triangle HUS is shown below.

If point G is located on US and HG is drawn, which additional information is sufficient to prove △HUG ≅ △HSG by SAS?

(1) HG bisects US
(2) HG is an altitude
(3) HG bisects ∠UHS
(4) HG is the perpendicular bisector of US.

Answer: (4) HG is the perpendicular bisector of US.

For SAS to apply, you would need two pairs of corresponding sides and the pair of angles between them to be congruent.

In Choice (1), HG bisecting US would give us two pairs of congruent sides, but it doesn't guarantee that the angles are congruent. Eliminate Choice (1).

In Choice (2), HG being an altitude would mean that each triangle would have a right angle and contain side HG, but we wouldn't know that SG is congruent to GU. Eliminate Choice (2).

In Choice (3), HG bisecting ∠UHS would mean that each triangle would have a congruent angle and contain side HG, but we wouldn't anything about sides HU and HS. Eliminate Choice (3).

In Choice (4), HG bisecting US perpendicularly would give us two pairs of congruent sides and a pair of right angles. So △HUG ≅ △HSG by SAS. Choice (4) is the correct answer.

11. The area of the base of a cone is 9π square inches. The volume of the cone is 36π cubic inches. What is the height of the cone in inches?

(1) 12
(2) 8
(3) 3
(4) 4

Answer: (1) 12

The Volume of a cone is 1/3 of the Area of the circular base times its height.

36π = (1/3) (9π) h
36π = (3π) h
12 = h

The correct answer is Choice (1).

If the problem was about a cylinder instead of a cone, then (4) would've been the answer.

12. On the set of axes below, AB, CD, EF, GH, and IJ are drawn.

Which segment is the image of AB after a dilation with a scale factor of 2 centered at (-2,-1)?

(1) CD
(2) EF
(3) GH
(4) IJ

Answer: (2) EF

Starting at point (-2,-1), you have to go 3 units up to get to (-2,2), which is a point on AB. If point (-2,2) is dilated with a scale factor of 2 centered on (-2,-1), it will be another 3 units up, which is the point at (-2,5), which is a point on line EF.

If you want more confirmation, consider that to get from (-2,1) to point B, you have to go 2 units up and 4 units to the right. Starting at point B, it you go another 2 units up and 4 units to the right, you end up at point F.

The correct answer is Choice (2).

13. Trapezoid ABCD is graphed on the set of axes below.

Which transformation would map point A onto A'(3,-7)?

(1) reflection over y = x
(2) reflection over the y-axis
(3) rotation of 180° about (0,0)
(4) rotation of 90° counterclockwise about (0,0)

Answer: (1) reflection over y = x

Point A is located at (-7,3) in Quadrant II. Point A' will be at (3,-7) is quadrant IV.

In Choice (1), a reflection over the line y = x would switch the x and y coordinates so that (-7,3) goes to (3,-7). This is the correct answer.

In Choice (2), a reflection over the y-axis would bring (-7,3) to (7,3). Eliminate Choice (2).

In Choice (3), a rotation of 180° about (0,0) would bring (-7,3) to (3,-7). Eliminate Choice (3).

In Choice (4), rotation of 90° counterclockwise about (0,0) would bring (-7,3) into Quadrant IV at point (-3,-7). Eliminate Choice (4).

14. A storage building is modeled below by a hemisphere on top of a cylinder. The diameter of both the cylinder and hemisphere is 12 feet. The total height of the storage building is 30 feet.

To the nearest cubic foot, what is the volume of the storage building?

(1) 942
(2) 2488
(3) 3167
(4) 3845

Answer: (3) 3167

If the diameter of the hemisphere is 12 feet, then the radius of the hemisphere is 6 feet. This means that the height of the cylinder is 30 - 6 = 24 feet. The volume of the building equals the volume of the cylinder plus the volume of the hemisphere.

V = πr²h + (1/2) (4/3) πr³
= π (6)² (24) + (2/3) π (6)³ = 3166.7

That's 3167 to the nearest cubic foot, which is Choice (3).

Side note: in Choice (4), you would get 3845 if you used 30 as the height of the cylinder instead of 24. I don't have a guess for how they came up with the numbers in Choices (1) and (2).

15. Which regular polygon will carry onto itself after a 135° rotation about its center?

(1) triangle
(2) pentagon
(3) hexagon
(4) octagon

Answer: (4) octagon

Which of the listed polygons have an exterior angle that is a factor of 135°?

A regular triangle is equilateral and has 360°/3 = 120° exterior angles, which is not a factor of 135°.

A regular pentagon has 360°/5 = 72° angles, which is not a factor of 135°.

A regular hexagon has 360°/6 = 60° angles, which is not a factor of 135°.

And a regular pentagon has 360°/8 = 45° angles, which is a factor of 135°.

Choice (4) is correct.

16. What is the length of the radius of the circle whose equation is x² + y² - 2x + 4y - 5 = 0?

(1) √(5)
(2) √(10)
(3) 5
(4) 10

Answer: (2) √(10)

Rearrange the terms and complete the squares to rewrite the equation in standard form.

x² + y² - 2x + 4y - 5 = 0
x² - 2x + y² + 4y - 5 = 0
x² - 2x + 1 + y² + 4y + 4 - 5 + 5 = 0 + 1 + 4 + 5
(x - 1)² + (y - 2)² = 10

So the square of the length of the radius, r² = 10. That means that r = √(10), which is Choice (2).

More to come. Comments and questions welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

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

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Geometry Regents

Part I

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


1. In right triangle LMN below, LN = 8, MN = 15, and LM = 17.
If triangle LMN is translated such that it maps onto triangle XYZ, which statement is always true?

(1) XY = 15
(2) YZ = 17
(3) m∠Z = 90°
(4) m∠X = 90°

Answer: (3) m∠Z = 90°

If LMN is translated on XYZ, then L maps to X, M maps to Y, and N maps to Z, and those pairs of angles are congruent. Also, LM maps to XY, MN maps to YZ, and LN maps to XZ.

Choice (1) says XY = 15, but LM = 17, so this is incorrect.

Choice (2) says YZ = 17, but MN = 15, so this is incorrect.

Choice (3) says Z is a right angle. N maps to Z, and N is a right angle, so this is true.

Choice (4) says X is a right angle, but L maps to X, and L is not a right angle, so this is incorrect.

Choice (3) is the correct choice.

2. Directed line segment KC has endpoints K(-4,-2) and C(1,8). Point E divides KC such that KE:EC is 3:2. What are the coordinates of point E?

(1) (-1,4)
(2) (-2,2)
(3) (-3,0)
(4) (0,6)

Answer: (1) (-1,4)

The x-coordinate of E will be 3/5ths of the way from -4 to 1, and the y-coordinate of E will be 3/5ths of the way from -2 to 8.

From K to C, the x-coordinate changes 1 - (-4) = 5 units, and 3/5 of 5 = 3. Add 3 to -4 to get -1 as the x-coordinate of E.

From K to C, the y-coordinate changes 8 - (-2) = 10 units, and 3/5 of 10 = 6. Add 6 to -2 to get 4 as the y-coordinate of E.

Point E is at (-1, 4), which is Choice (1).

3. In right triangle DAN, m∠A = 90°. Which statement must always be true?

(1) cos D = cos N
(2) cos D = sin N
(3) sin A = cos N
(4) cos A = tan N

Answer: (2) cos D = sin N

If A is the right angles, then the sin D = cos N and sin N = cos D.

Choice (2) is the correct answer.

4. In the diagram below of parallelogram RSTV, diagonals SV and RT intersect at E.

Which statement is always true?

(1) SR ≅ RV
(2) RT ≅ SV
(3) SE ≅ RE
(4) RE ≅ TE

Answer: (4) RE ≅ TE

The diagonals of a parallelogram bisect each other but they are not congruent to each other. So SE ≅ VE and RE ≅ TE.

Choice (1) has consecutive sides of a parallelogram, which are only congruent in rhombuses.

Choice (2) has congruent diagonals, which is only true in rectangles.

Choice (3) says that half of each diagonal must be congruent, but that is only true in rectangles.

Choice (4) says that the two halves of RT are congruent to each other, which is always true. Choice (4) is the correct answer.

5. In △SNA below, UE || NA.

If SU = 3, SN = 11, and EA = 13, what is the length of SE, to the nearest tenth?

(1) 2.5
(2) 3.5
(3) 4.9
(4) 17.9

Answer: (3) 4.9

Because UE || NA, SU / UN = SE / EA, so 3/8 = SE / 13.

Therefore, 8 SE = 39, and SE = 39/8, which is approximately 4.9. This is Choice (3).

6. Many roofs are slanted to prevent the buildup of snow. As modeled below, the length of a roof is 5.5 meters and it rises to a height of 2.5 meters

The angle of elevation of the roof, to the nearest degree, is

(1) 24°
(2) 25°
(3) 27°
(4) 28°

Answer: (3) 27°

You are given the hypotenuse of the triangle and the side that is opposite the angle of elevation. Opposite and hypotenuse means that you need to use the sine ratio.

sin x = 2.5 / 5.5
x = sin^-1(2.5/5.5) = 27.03..., which is 27 to the nearest degree.

The correct answer is Choice (3).

As a side note, I'm surpised that they didn't include 63 degrees, which would be the result is you accidentally used sine. If you used tangent, you would have gotten 24.4, which rounds down to 24 or (mistakely) up to 25.

7. In the diagram below, CT || AR , and ACE and RC are drawn such that AC ≅ RC

If m∠ECT = 75°, what is m∠ACR?

(1) 30°
(2) 60°
(3) 75°
(4) 105°

Answer: (1) 30°

Triangle CAR is an isosceles triangle, so the base angles, CAR and ARC, are congruent. Angles ECT and CAR are corresponding angles, which are also congruent.

If angle ECT has a measure of 75 degrees, then CAR and ARC are each 75 degrees. This means that the measure of ACR, the third angle of triangle CAR, has a measure if 180 - (75 + 75) = 30 degrees.

This is Choice (1).

8. In the diagram below, △ABC has medians AX, BY, and CZ that intersect at point P.
If AB = 26, AC = 28, and PC = 16, what is the perimeter of △CZA?

(1) 57
(2) 65
(3) 70
(4) 73

Answer: (2) 65

Medians are drawn to midpoints of the opposite sides, and the midpoint divides a line segment into two equal, congruent segments. Medians meet at a point called the centrod which is 2/3 of the way from the angle to the midpoint.

The perimeter of CZA is the sum of the sides CZ, ZA and AC.

CZ is the sum of CP + PZ, and CP is twice the length of PZ, so CZ = 16 + 8 = 24.

ZA is half the length of AB, which is 26, so ZA = 13.

AC is given to be 28.

Therefore, the perimeter of CZA is 24 + 13 + 28 = 65, which is Choice (2).

More to come. Comments and questions welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!