(x, why?): April 2024

Monday, April 29, 2024

January 2024 Algebra 2 Regents, Part I (multiple choice)

This exam was adminstered in January 2024.

January 2024 Algebra 2 Regents

Part I

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

17. Which equation does not represent an identity?

(1) x² - y² = (x + y)(x - y)
(2) (x - y)² = (x - y)(x - y)
(3) (x + y)² = x² + 2xy + y²
(4) (x + y)³ = x³ + 3xy + y³

Answer: (4) (x + y)³ = x³ + 3xy + y²

Choice (4) does not show the expansion of a binomial raised to the third power. The exponents for each term should add up to 3, but the middle term is 3xy, with two exponents of 1.

Choice (1) is the Difference of Squares rule. Eliminate Choice (1).

Choice (2) is the definition of squaring a binomial (or squaring anything). Eliminate Choice (2).

Choice (3) is the expansion of a binomial squared: x² + xy + xy + y². Eliminate Choice (3).

The third line of Pascal's Triangle is 1 3 3 1, and the expansion of (x + y)³ is x³ + 3x²y + 3xy² + y³.

The correct answer is Choice (4).

18. Two surveys were conducted to estimate the proportion of teens who use social media at least once per day.

Based on these results, it was determined that approximately 75% of teens use social media at least once per day. What is the best explanation of the difference in the results between the two surveys?

(1) The smaller sample size of five teens resulted in a smaller margin of error and should provide a more accurate estimate.
(2) The smaller sample size of five teens resulted in a bigger margin of error and should provide a more accurate estimate.
(3) The larger sample size of 50 teens resulted in a smaller margin of error and should provide a more accurate estimate.
(4) The larger sample size of 50 teens resulted in a bigger margin of error and should provide a more accurate estimate.

Answer: (3) The larger sample size of 50 teens resulted in a smaller margin of error and should provide a more accurate estimate.

A larger sample size means that the results will be more accurate and result is a smaller margin of error.

Five students is too small a sample size to be representative of the entire population.

19. Given f(x) = x³ - 3 and f^-1 = ∛(x - 3b), the value of b is

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

Answer: (2) -1

The definition of inverse functions is that f^-1(f(x)) = x. The inverse undoes the function and yields the original input.

You can plug in the choices and see which one works, or you can rewrite the equation.

y = x³ - 3

x = y³ - 3

x + 3 = y³

∛(x + 3) = y

So if 3 = -3b, then b = -1

The correct Choice is (2).

20. Robert is buying a car that costs $22,000. After a down payment of $4000, he borrows the remainder from a bank, a six year loan at 6.24% annual interest rate. The following formula can be used to calculate his monthly loan payment. R = ((P)(i)) / (1 - (1 + i)^-t
R = monthly payment
P = loan amount
i = monthly interest rate
t = time, in months
Robert’s monthly payment will be

(1) $298.31
(2) $300.36
(3) $307.35
(4) $367.10

Answer: (2) $300.36

Make the substitutions and put the expression into your calculator.

Note that the amount borrowed is 22000 - 4000 = 18000, 6.24% is the annual rate, which must be divided by 12, and the number of months is 6 years times 12 months / year, or 72.

R = ( (18000)(0.0624/12) ) / (1 - (1 + 0.0625/12)^-72

R = 300.355 or $300.36, which is Choice (2).

21. Given tan θ = -4/3 where π/2 < θ < π, what is the value of sec θ?

(1) -5/3
(2) -3/5
(3) 4/5
(4) 5/3

Answer: (1) -5/3

Given that tan = sin / cos and sec = 1 / cos, then sec is either 5/3 or -5/3 depending upon the value of θ, which is this question put the point in Quadrant II where cosine is negative.

So the correct answer is Choice (1) -5/3.

22. To solve the equation 7 / (x + 7) + 4x / (x - 7) = (3x + 7) / (x - 7), Joan’s first step is to multiply both sides by the least common denominator. Which statement is true?

(1) -14 is an extraneous solution.
(2) 7 and -7 are extraneous solutions.
(3) 7 is extraneous solutions.
(4) There are no extraneous solutions.

Answer: (3) 7 is extraneous solutions.

The equation is not defined for x = 7 or x = -7. If you multiply the equation by the least common demoninator, you need to multiple by (x + 7)(x - 7), which means that there will be two possible extraneous solutions. However, it's possible that either one, or both, are NOT solutions at all.

7 / (x + 7) + 4x / (x - 7) = (3x + 7) / (x - 7)

7 (x - 7) + 4x (x + 7) = (3x + 7)(x + 7)

7x - 49 + 4x² + 28x = 3x² + 28x + 49

x² + 7x - 98 = 0

(x + 14)(x - 7) = 0

x = -14 or x = 7

Only 7 is an extraneous solution.

23. Beginning July 1, 2019, Michelle deposited $250 into an account that yields 0.15% each month. She continued to make $250 deposits into this account on the first of each month for 3 years. Which expression represents the amount of money that was in the account after her last deposit was made on June 1, 2022?

(1) 250(1.0015)³
(2) 250(1.0015)³⁶
(3) (250 - 250(1.0015)³) / (1 - 1.0015)
(4) (250 - 250(1.0015)³⁶) / (1 - 1.0015)

Answer: (4) (250 - 250(1.0015)³⁶) / (1 - 1.0015)

Choices (1) and (2) are compound interest with no extra deposits. Eliminate Choices (1) and (2).

Choice (3) only has 3 months of interest. Eliminate Choice (3).

Choice (4) is the correct geometric series and the correct choice.

24. A study of the red tailed hawk population in a given area shows the population, H(t), can be represented by the function H(t) = 50(1.19)^t where t represents the number of years since the study began. In terms of the monthly rate of growth, the population can be best approximated by the function

(1) H(t) = 50(1.015)^12t
(2) H(t) = 50(1.15)^t/12
(3) H(t) = 50(1.19)^12t
(4) H(t) = 50(1.19)^t/12

Answer: (1) H(t) = 50(1.015)^12t

The annual rate needs to be converted to the monthly rate.

50(1.19^1/12)^12t = 50(1.0146...)^12t

This is the same as Choice (1), which is the correct answer.

More to come.

Comments and questions welcome.

More Regents problems.

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Sunday, April 28, 2024

You Have to Spend Money

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

Not the same thing at all.

Today's logic lesson: If someone tells you that you have to spend money to make money, don't for one moment believe that they're saying that if you spend money then you will make money. Those statements are not logically equivalent.

If the former case, you can spend money and not make money. The first statement doesn't say that you actually will make money if you spend money, only that you won't make money if you do not. This statement is only false if you make money without spending any.

What they'd like you to hear is the second statement, which is a more traditional p then q statement. In this instance, you may or may not make money without spending money. On the other hand, the statement is false if you spend money and don't make money.

Many poor planners or investors will tell you that it's quite possible that you can spend money and not make money. You hope and plan for a better outcome.

Speaking of spending money and making money: if you like reading fantasy and science fiction, consider spending money on any of the books advertised below, three of which I wrote, and one where my stories are included along with 12 other authors. Will you make money if you spend it this way? No, but I might make a few pennies. And isn't that worth it? ... Anyone? ... Anyone? ...

I also write Fiction!

And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!

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Come back often for more funny math and geeky comics.

January 2024 Algebra 2 Regents, Part I (multiple choice)

This exam was adminstered in January 2024.

More Regents problems.

January 2024 Algebra 2 Regents

Part I

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

9. The equation of the parabola that has its focus at the point (-3, 2) and directrix at y = 0 is

(1) y = 1/4 (x + 3)² + 1
(2) y = 1/4 (x - 3)² + 1
(3) y = 1/8 (x + 3)² + 1
(4) y = 1/8 (x - 3)² + 1

Answer: (1) y = 1/4 (x + 3)² + 1

If the focus (which is on the axis of symmetry) is on x = -3, then the sign is flipped in the equation. Eliminate Choices (2) and (4).

The directrix of the parabola is y = k - a = 0, so k = a. And since the vertex is the midpoint between the focus and the directrix, (2 + 0)/2 = 1, so k = a = 1.

The equation of the parabola would be written as

y = (1/4(1))(x + 3)² + 1

This is Choice (1).

10. The seventh term of the geometric sequence √(6), -2√(3), 2√(6), -4√(3), ... is

(1) 6√(6)
(2) -6√(3)
(3) 8√(6)
(4) -8√(3)

Answer: (3) 8√(6)

If you divide 2√(6) by -2√(3), you will get a common ratio of -√(2),

There is no reason to use a formula. Just multiply by -√(2) three more times.

4√(6), -8√(3), 8√(6).

The answer is Choice (3).

11. A company wishes to determine the cooking time for one pound of spaghetti. The company’s technicians cooked one pound of spaghetti and recorded the time needed for the spaghetti to be ready to eat. Repeating this process 35 times resulted in an approximately normal distribution, with a mean of 9.82 minutes and a standard deviation of 1.4 minutes. In which interval should the middle 95% of cooking times fall?

(1) (8.42, 11.22)
(2) (7.02, 12.62)
(3) (9.35, 10.29)
(4) (6.82, 11.32)

Answer: (2) (7.02, 12.62)

The middle 95% interval is 9.82 + 2(1.4), which 9.82 - 2.8 = 7.02 and 9.82 + 2.8 = 12.62.

The correct answer is Choice (2).

If you forgot to multiply by 2, you would've gotten Choice (1).

12. Given f(x) = 2x² + 7x - 15 and g(x) = 3 - 2x, what is f(x)/g(x) for all defined values?

(1) -x - 5
(2) -x + 5
(3) x - 5
(4) x + 5

Answer: (1) -x - 5

Divide 2x² + 7x - 15 by -2x + 3. It's as simple as factoring.

2x² + 7x - 15

(-2x + 3)(-x - 5)

Notice that 2x² divided by -2x is -x, and -15 divided by 3 is -5. You can check -2x * -5 = 10x and 3 * -x = -3x, and 10 - 3x = 7x. Check.

Choice (1) is the correct answer.

13. Which equation is equivalent to P = 210x^4/3y^7/3

Answer: (3) [See Image]

Change the improper fractions into mixed number exponents. Separate the whole number portion from the fraction. The fractions both have denominators of 3 meaning that it is a cube root. Only the variables with fractional exponents will go under the cube root sign.

210x^4/3y^7/3 = 210 x¹ x^1/3 y² y^1/3

= 210 xy² x^1/3 y^1/3

= 210 xy² ∛(xy)

This is Choice (3).

14. The average cost of a gallon of milk in the United States between the years of 1995 and 2018 can be modeled by the equation P(t) = -0.0004t³ + 0.0114t² - 0.0150t + 2.6602, where P(t) represents the cost, in dollars, and t is time in years since January 1995. During this time period, in what year did P(t) reach its maximum?

(1) 1995
(2) 2013
(3) 2014
(4) 2018

Answer: (2) 2013

Graph the equation and look at the table of values to find the maximum. You'll see it occurs about year 18.

Add 18 years to 1995 to get 2013.

15. The temperature, F, in degrees Fahrenheit, after t hours of a roast put into an oven is given by the equation F = 325 - 185e^-0.4t. What was the temperature of the roast when it was put into the oven?

(1) 325
(2) 200
(3) 185
(4) 140

Answer: (4) 140

When t = 0, the exponent will be zero, so e⁰ = 1 and 185e⁰ = 185.

Subtract F = 325 - 185 = 140, which is Choice (4).

16. The roots of the equation 0 = x² + 6x + 10 in simplest a + bi form are

(1) -3 + 2i
(2) -6 + i
(3) -3 + i
(4) -3 + i√(2)

Answer: (3) -3 + i

Find the discriminant using b² - 4ac (don't confuse the a's and b's here).

6² - 4(1)(10) = 36 - 40 = -4. And the square root of -4 is 2i.

So the roots are x = (-6 + 2i) / (2(1)), which simplifies to -3 + i.

You could also Complete the Square:

x² + 6x + 10 = 0

x² + 6x + 9 = -1

(x + 3)² = -1

x + 3 = + i

x = -3 + i

The correct answer is Choice (3).

More to come.

Comments and questions welcome.

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I also write Fiction!

And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!

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Friday, April 26, 2024

January 2024 Algebra 2 Regents, Part I (multiple choice)

This exam was adminstered in January 2024.

More Regents problems.

January 2024 Algebra 2 Regents

Part I

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

1. A cafeteria food manager studied the lunchtime eating habits of a group of employees in their office building. The purpose of the study was to determine the proportion of employees who purchased lunch in the cafeteria, brought their lunch from home, or purchased lunch from an outside vendor. This collection of data would best be classified as

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

Answer: (3) an observational study

This is an example of an observational study. There was no experiment or simulation conducted. No interviews were made.

2. Which graph has imaginary roots?

Answer: (2) [See Image]

The roots of a quadratic equation are the x-intercepts of the graph, the points were y = 0.

If the quadratic function does not cross the x-axis then it does not have any real roots and only has imaginary roots. Choice (2) is the only image that doesn't show the graph crossing the x-axis.

3. Given 3 is a root of f(x) = x⁴ − x³ − 21x² + 45x, what are the other unique roots of f(x)?

(1) −5, only
(2) −5 and 0
(3) −3, 1 and 5
(4) −5, −3 and 0

Answer: (2) −5 and 0

You can graph the equation in your calculator and check the table of value to see which inputs give a value of zero.

Note that since every term has a variable in it, then f(0) will equal 0, so Choices (1) and (3) can be eliminated.

If you check x = -3, f(-3) = -216, so Choice (4) can be eliminated.

Choice (2) is the correct answer.

4. Given p ≠ q, p = (1/2)^q, expressed in logarithmic form, is equivalent to

(1) log_p (1/2) = q
(2) log_q (p) = 1/2
(3) log_1/2 (p) = q
(4) log_p (q) = p

Answer: (3) log_1/2 (p) = q

Rewrite the equation using 1/2 as the base.

p = (1/2)^q

log _1/2 p = log _1/2 (1/2)^q

log _1/2 p = q

Choice (3) is the correct answer.

5. Which graph best represents the graph of f(x) = (x + a)²(x − b), where a and b are positive real numbers?

Answer: (1) [See Image]

In a cubic equations without a negative leading coefficient, the graph starts at negative infinity when x is negative and goes to positive infinity when x is positive. Eliminate Choices (3) and (4).

Since both a and b are positive numbers, then the roots of the equation will occur at x = -a and x = b. These are the only two roots. Therefore, Choice (2) can be eliminated because it includes x = 0 as a root.

Moreover, since (x + a) has a exponent of 2, then x = -a will be an inflection point where the graph will decrease again. This is the case in Choice (1), which is the correct answer.

6. The equations y = 3t + 6 and y = (1.82)^t approximately model the growth of two separate populations where t > 0. What is the best approximation of the time, t, at which the populations are the same?

(1) -1.9
(2) 0.3
(3) 5.1
(4) 21.3

Answer: (2) x − 3 is a factor of p(x)

Graph the two equations. You will find that the two eqautions will intersect at approximately t = -1.9 and t = 5.1.

Since t represents time, we can discard the negative outcome, and accept t = 5.1, which is Choice (3).

Note that 0.3 and 21.3 are the y values you will when x = -1.0 and x = 5.1, respecitvely.

7. Given y = -2x and x² + y² = 5, the point of intersection in Quadrant II is

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

Answer: (4) (-1,2)

The points in Quadrant II have a negative x coordinate and a positive y coordinate. Eliminate Choice (1).

If y = -2x, then y will be a positive number when x is an integer. Eliminate Choices (2) and (3).

In Choice (4), 2 = -2(-1) and (-1)² + (2)² = 5. Choice (4) is correct.

8. The rational expression (2x⁴ − 5x² + 3x − 2) / (x - 3) is equivalent to

(1) 2x³ − 5x − 12 − 38/(x - 3)
(2) 2x³ + 6x² + 13x + 42 + 124/(x - 3)
(3) 2x³ − 5x + 18 − 56/(x - 3)
(4) 2x³ - 6x² + 13x − 36 − 106/(x - 3)

Answer: (2) 2x³ + 6x² + 13x + 42 + 124/(x - 3)

Long division will give you the answer.

More to come.

Comments and questions welcome.

More Regents problems.

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Thursday, April 25, 2024

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

This exam was adminstered in January 2024.

More Regents problems.

January 2024 Geometry Regents

Part I

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

17. In parallelogram ABCD with AC ⟂ BD, AC = 12 and BD = 16. What is the perimeter of ABCD?

(1) 10
(2) 24
(3) 40
(4) 56

Answer: (3) 40

The parallelogram is a rhombus because the diagonals are perpendicular. The diagonals form four right triangles with legs of 6 and 8 (which is half the size of 12 and 16, respectively.)

Using Pythagorean Theorem (or just knowing about 6-8-10 right triangles), you will find that each side of the rhombus has a length of 10. This means that the perimeter is 10 * 4 = 40, which is Choice (3).

I'm surprised that neither 20 nor 80 were listed as incorrect choices.

18.In the diagram of △CAT below, m∠A = 90° and altitude AE is drawn from vertex A.

Which statement is always true?

(1) CE / AE = AE / ET
(2) AE / CE = AE / ET
(3) AC / CE = AT / ET
(4) CE / AC = AC / ET

Answer: (1) CE / AE = AE / ET

When an altitude is drawn from the hypotenuse to the right angle of a right triangle, two more right triangles are formed and all three right triangles are similar to each other. That means that the corresponding sides of any two of the triangles are proportional.

Choice (1) is the Right Triangle Altitude Theorem. It is the correct choice.

Choice (2) shows long leg / short leg = short leg / long leg. This is incorrect.

Choice (3) shows hypotenuse / short leg = hypotenuse / long leg. This is incorrect.

Choice (4) has a ratio that contains two legs from different triangles. This is incorrect.

19. A sandbox in the shape of a rectangular prism has a length of 43 inches and a width of 30 inches. Jack uses bags of sand to fill the sandbox to a depth of 9 inches. Each bag of sand has a volume of 0.5 cubic foot. What is the minimum number of bags of sand that must be purchased to fill the sandbox?

(1) 14
(2) 13
(3) 7
(4) 4

Answer: (1) 14

Find the volume, convert it to cubic feet, and then divide it by 0.5 to find the number of bags.

V = l * w * h = 43 * 30 * 9 = 11610, 11610 / (12^3) = 6.71875 cu. ft.

6.71875 / .5 = 13.4375

You would need 14 bags to fill the sandbox. (Note: 13 bags wouldn't fill the sandbox, so don't round down. Round up.)

20.Parallelogram EATK has diagonals ET and AK. Which information is always sufficient to prove EATK is a rhombus?

(1) EA ⟂ AT
(2) EA ≅ AT
(3) ET ≅ AK
(4) ET ≅ AT

Answer: (2) EA ≅ AT

It is a rhombus is the diagonals are perpedicular or the consecutive sides are congruent.

In Choice (1), EA and AT are not diagonals. This makes the shape a rectangle.

In Choice (2), EA and AT are consecutive sides. This makes the shape a rhombus. Choice (2) is the correct answer.

In Choice (3), ET and AK are congruent diagonals. This makes the shape a rectangle.

In Choice (4), ET is a diagonal and AT is a side. Being congruent means nothing. It's just a parallelogram.

21. In the diagram below, ABCD || EHK, and MBHP and NCHL are drawn such that BC ≅ BH.

If m∠NCD = 62°, what is m∠PHK?

(1) 118°
(2) 68°
(3) 62°
(4) 56°

Answer: (4) 56°

Because BC ≅ BH, then BCD is an isosceles triangle.

This means that ∠BCH = 62, ∠BHC = 62 and ∠CBH = 56.

Since the lines are parallel, ∠CBH and ∠PHK are corresponding angles, which must be congruent.

The measure of angle PHK is 56 degrees.

22. Triangles YEG and POM are two distinct non-right triangles such that ∠G > ∠M. Which statement is sufficient to prove △YEG is always congruent to △POM?

(1) ∠E ≅ ∠O and ∠Y ≅ ∠P
(2) YG ≅ PM and YE ≅ PO
(3) There is a sequence of rigid motions that maps ∠E onto ∠O and YE onto PO.
(4) There is a sequence of rigid motions that maps point Y onto point P and YG onto PM.

Answer: (3) There is a sequence of rigid motions that maps ∠E onto ∠O and YE onto PO.

Make a little sketch, so it'll be easier to compare.

Choice (1) would just show that the two triangles are similar, but not congruent. Eliminate Choice (1).

Choice (2) shows SSA, which is not sufficient for congruency. The question states that they are not right triangles, so it HL Theorem doesn't apply either. Eliminate Choice (2).

Choice (3) shows AAS, which is sufficient to show that two triangles are congruent. Choice (3) is the correct answer.

Choice (4) has one side and one angle. This is insufficient for congruency. Eliminate Choice (4).

23.In the diagram of triangles ABD and CBE below, sides AD and CE intersect at F, and ∠ADB ≅ ∠CEB.

Which statement can not be proven?

(1) △ADB ≅ △CEB
(2) ∠EAF ≅ ∠DCF
(3) △ADB ~ △CEB
(4) △EAF ~ △DCF

Answer: (1) △ADB ≅ △CEB

There is not enough information to show that the two triangles are congruent. Note that no line segements are said to be either congruent or even perpendicular.

Choice (1) is not possible.

Choice (2) can be proven because CEA is congruent to ADC because they are supplementary to the congruent angles. Angles AFE and CFD are congruent because they are vertical angles. If two pairs of angles in two triangles are congruent, then the third pair must be congruent. So angle EAF is congruent to angle DCF.

Choice (3) can be proven because ∠B is congruent to itself by the Reflexive Property, and we were given ∠ADB ≅ ∠CEB. By AA, the two triangles are similar.

Choice (4) can be proven because ∠ADC ≅ ∠CEA because they are supplementary to the given congruent angles. Angles AFE and CFD are congruent because they are vertical angles. By AA, the two triangles are similar.

24. A small town is installing a water storage tank in the shape of a cylinder. The tank must be able to hold at least 100,000 gallons of water. The tank must have a height of exactly 30 feet. [1 cubic foot holds 7.48 gallons of water]
What should the minimum diameter of the tank be, to the nearest foot?

(1) 12
(2) 24
(3) 65
(4) 75

Answer: (2) 24

Figure out the volume from the number of gallons. V = pi r²h. Since you know the height, you can find the radius. Double that to get the diameter.

100,000 gallons * 1 cu ft / 7.48 gallons = 13368.984 cu ft

V = π r² h

13368.984 = π r² (30)

r² = 13368.984 / (30π) = 141.85

r = 11.9

To the nearest foot, the minimum diameter is 24 feet, which is Choice (2).

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!

And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!

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Wednesday, April 24, 2024

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

This exam was adminstered in January 2024.

More Regents problems.

January 2024 Geometry Regents

Part I

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

9. In the diagram below, △GHJ is dilated by a scale factor of 1/2 centered at point B to map onto △CDF.

If m∠DFC = 40°, what is m∠HJG?

(1) 20°
(2) 40°
(3) 60°
(4) 80°

Answer: (2) 40°

A dilation preserves the shape of the original object, so the corresponding angles are congruent. Since ∠DFC and ∠HJG are corrsponding, they are congruent.

The correct answer is Choice (2).

10. Directed line segment AJ has endpoints whose coordinates are A(5,7) and J(-10,-8). Point E is on AJ such that AE:EJ is 2:3. What are the coordinates of point E?

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

Answer: (4) (-1,1)

AE is 2/5th of the segment AJ and EJ is 3/5th of the segment.

The change is x coordinates is -15. Multiply (-15)(2/5) and you get -6. Subtract 6 from 5, and the x-coordinate is -1. Choice (4) is the only possible answer.

The change in y coordinates is also -6. Subtract 6 from 7 and you get 1 as the y-coordinate of E. E is (-1,1), which is Choice (4).

11. A tipping platform is a ramp used to unload trucks, as shown in the diagram below.

The truck is on a 75-foot-long ramp. The ramp is tipped at an angle of 30°. What is the height of the upper end of the ramp, x, to the nearest tenth of a foot?

(1) 68.7
(2) 65.0
(3) 43.3
(4) 37.5

Answer: (4) 37.5

The ramp is the hypotenuse and the height is opposite the angle. That means you need to use the sine ratio.

Sin 30 = x / 75

x = 75 sin 30 = 37.5, which is Choice (4).

The sine of 30 degrees is .5, so it is half the size of the rampl.

12. In the diagram below of right triangle MET, altitude ES is drawn to hypotenuse MT.

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?
If ME = 6 and SM = 4, what is MT?

(1) 9
(2) 8
(3) 5
(4) 4

Answer: (1) 9

Right triangles with altitudes drawn have three similar right triangles which have proportional sides.

So SM/ME = ME/MT and 4/6 = 6/MT

Cross-multiply and 4 MT = 36 and MT = 9, which is Choice (1).

Always reread the question: ST = 5, but that isn't what they asked.

13. In the diagram below of square CASH, diagonals AH and CS intersect at Z.

Which statement is true?

(1) m∠ACZ > m∠ZCH
(2) m∠ACZ < m∠ASZ
(3) m∠AZC = m∠SHC
(4) m∠AZC = m∠ZCH

Answer: (3) m∠AZC = m∠SHC

The four corner angles of a square are right angles, which are 90 degrees. The diagonals of a square are perpendicular and meet at right angles, which are 90 degrees. The diagonals bisect the corner right angles, creating 45 degree angles. So every angle in the image is either 45 degrees or 90 degrees.

In Choice (1), both angles are 45 degrees. They are equal so eliminate Choice (1).

In Choice (2), both angles are 45 degrees. They are equal so eliminate Choice (2).

In Choice (3), both angles are 90 degrees. Choice (3) is the correct choice.

In Choice (4), angle AZC is 90 degrees but ZHC is 45 degrees. They are not equal so eliminate Choice (4).

14. In the diagram below of circle O, secants CFD and CHE are drawn from external point C.

If m DE = 136° and m∠C = 44°, then m FH is

(1) 46°
(2) 48°
(3) 68°
(4) 88°

Answer: (2) 48°

The measure of angle C is equal to one-half the difference of the two arcs it intercepts.

That is, C = 1/2 (DE - FH), so

44 = 1/2 (136 - FH)

88 = 136 - FH

FH = 136 - 88 = 48 degrees

The answer is Choice (2).

15. A right circular cylinder has a diameter of 8 inches and a height of 12 inches. Which two-dimensional figure shows a cross section that is perpendicular to the base and passes through the center of the base?

Answer: (4) [See Image]

A cross section that is perpendicular to the base of a cylinder will have the shape of a rectangle.

Since is goes through the center of the base, it width of the rectangle will be the diameter of the cylinder. The length of the rectangle will be the height of the cylinder. The only possible choice is Choice (4).

If the cross section was parallel to the base, then Choice (1) would have been the correct choice because a radius of 4 means a diameter of 8.

16. On the set of axes below, AB is drawn and passes through A(-2,6) and B(4,0).

If CD is the image of AB after a dilation with a scale factor of 1/2 centered at the origin, which equation represents CD?

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

Answer: (2) y = -x + 2

A dilation of a line will create an image that is parallel to the original line (or it will be the same as the original line if the center of dialation is on the line.)

Since it is parallel, the slope will not change. The slope of the given line is -1, which can be seen from the rise and the run. Eliminate Choices (3) and (4).

The original line has a y-intercept of +4, which is 4 units away from the origin. The image would be 1/2 as far away from the origin, which means that the y-intercept of the image will be at +2. This is Choice (2).

More to come. Comments and questions welcome.

More Regents problems.

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Tuesday, April 16, 2024

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

This exam was adminstered in January 2024.

More Regents problems.

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.

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!

And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!

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