August 2023 Algebra 2, Part II

This exam was adminstered in August 2023.

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Algebra 2 August 2023

Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


25. Factor the expression 2x³ - 3x² - 18x + 27 completely.

Answer:

Factor by grouping, and then factor the quadratic you get after the first step.

There are two ways to group, and either should work in any question of this kind.

2x³ - 3x² - 18x + 27
(2x³ - 3x²) - (18x - 27)
x²(2x - 3) - 9(2x - 3)
(x² - 9)(2x - 3)
(x + 3)(x - 3)(2x - 3)

You can also switch the two middle terms around. This is just the way I learned it, so I usually do it, especially if it helps me avoid factoring out a minus sign.

2x³ - 18x - 3x² + 27
(2x³ - 18x)(3x² - 27)
2x(x² - 9) - 3(x² - 9)
(2x - 3)(x² - 9)
(2x - 3)(x + 3)(x - 3)

26. Algebraically determine the values of x that satisfy the system of equations shown below:

y = x² + 8x - 5
y = 8x - 4

Answer:

Set the two statements equal to each other. Solve the quadratic equation for x. You don't need to find y.

y = x² + 8x - 5
y = 8x - 4

x² + 8x - 5 = 8x - 4
x² - 1 = 0
(x - 1)(x + 1) = 0
x - 1 = 0 or x + 1 = 0
x = 1 or x = -1

27. Solve the equation 3x² + 5x + 8 = 0. Write your solution in a + bi form.

Answer:

When they tell you "a + bi" form, you know that it won't be something that you can factor easily and that you need to use the quadratic formula.

3x² + 5x + 8 = 0

x = (-b + √(b² - 4ac) ) / (2a)

x = (-5 + √(5² - 4(3)(8)) ) / (2(3))

x = (-5 + √(25 - 96) ) / (6))

x = (-5 + √(-71) ) / (6))

x = -5/6 + i √(71)/6

28. On the coordinate plane below, sketch at least one cycle of a cosine function with a midline at y = -2, an amplitude of 3, and a period of π/2.

Answer:

Cosine doesn't start at the midline. Cosine starts at the midline plus the amplitude, which in this case is -2 + 3 = 1. The minumim will occur at y = -2 - 3 = -5 when x = (1/2)(π/2) = π/4.

Your sketch should look like this:

Remember that a sketch doesn't have to be perfect, but it shouldn't have any obvious errors, such as crossing the axis at the wrong place.

29. Given i is the imaginary unit, simplify (5xi³ - 4i)² as a polynomial in standard form.

Answer:

Use FOIL, the Distributive Property or the Box Method/Area Model. Remember that i² = -1, i³ = -i, i⁴ = 1, i⁵ = i, and so forth.

(5xi³ - 4i)²
25x²i⁶ - 40xi⁴ + 16i²
-25x² - 40x - 16

30. Consider the parabola given by y = 1/4 x² + x + 8 vertex (-2,7) and focus (-2,8). Use this information to explain how to determine the equation of the directrix.

Answer:

The focus is 1 unit above the vertex, so the directrix is the horizontal line that contains the point 1 unit below the vertex, (-2,6).

The equation would be y = 6.

Remember that you have to give an explanation, not just the equation. The equation alone is half-credit.

31.Write (x √(x³) ) / (∛(x⁵)

Answer:

Remove the radicals by rewriting the expressions with fractional exponents. Then using the laws of exponents subtract the exponent of the numerator from the exponent of the denominator.

(x √(x³) ) / (∛(x⁵)

x (x ^(3/2)) / x ^(5/3)

x ^(5/2) / x ^(5/3)

x ^{(5/2 - 5/3)}

x ^{(15/6 - 10/6)}

x ^(5/6)

32. A fruit fly population can be modeled by the equation P = 10(1.27)^t, where P represents the number of fruit flies after t days. What is the average rate of change of the population, rounded to the nearest hundredth, over the interval [0,10.5]? Include appropriate units in your answer.

Answer:

Find P(0) and P(10.5), subtract the latter from the initial value and then divide by 10.5.

( 10(1.27)^10.5 - 10(1.27)⁰) / (10.5 - 0) = 10.7628...

10.76 fruit flies per day.

End of Part II

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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.

June 2023 Algebra 2 Regents Part IV

This exam was adminstered in June 2023.

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Algebra 2 June 2023

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. The volume of air in an average lung during breathing can be modeled by the graph below.

Using the graph, write an equation for N(t), in the form N(t) = A sin (Bt) + C.

That same lung, when engaged in exercise, has a volume that can be modeled by E(t) = 2000 sin(πt) + 3200, where E(t) is volume in mL and t is time in seconds.

Graph at least one cycle of E(t) on the same grid as N(t).

How many times during the 5-second interval will N(t) = E(t)?

Answer:

A is the height of the graph from the midline, which is 400. B is the period which is 2π/5 because it takes 5 seconds before 2 pi is graphed. C is the midline, which is 2400. Since the graph starts at the midline, it is a sine function, not a cosine function.

The equation will be T = (400 - 75)e^-.0735t + 75.

For the second part, look at the graph below.

In a 5-second interval, N(t) = E(t) four times.

End of Exam

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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.

June 2023 Algebra 2 Regents, Part III

This exam was adminstered in June 2023.

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Algebra 2 June 2023

Part III: Each correct answer will receive 4 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.


33. Patricia creates a cubic polynomial function, p(x), with a leading coefficient of 1. The zeros of the function are 2, 3, and -6. Write an equation for p(x).

Sketch y = p(x) on the set of axes below.

Answer:

You are give the zeroes of the function and you need to have a 1 leading coefficient. You can write the function in factored form. You do not need to multiply it.

p(x) = (x - 2)(x - 3)(x + 6)

If you aren't sure what this looks like, you can put it in your graphing calculator and look at it and the table of values.

When a cubic function has a positive leading coefficient, it starts at negative infinity and climbs to positive infinity. The end behavior is that as x goes to negative infinity, y goes to negative infinity, and when x goes to positive infinity, y goes to positive infinity.

You also have to have the graph cross the x-axis three times, at x = 2, x = 3 and x = -6.

Your graph should look like this:

34. A public radio station held a fund-raiser. The table below summarizes the donor category and method of donation.

To the nearest thousandth, find the probability that a randomly selected donor was categorized as a supporter, given that the donation was made online.

Do these data indicate that being a supporter is independent of donating online? Justify your answer.

Answer:

There were 3216 donations made onlie. Of those, 1200 were from supporters. Probability is 1200/3216 = 0.3731..., or 0.373.

There were a total of 1600 supporters out of 4286 donations made, and 1600/4286 = 0.373. So, yes, it is indepenpent because the probability of being a supporter is equal to the probability of being a supporter who donated online.

35. Algebraically solve the system:
(x - 2)² + (y - 3)² = 20
y = -2x + 7

Answer:

The first equation is a circle and the second circle is a line. There may be 0, 1, or 2 possible solutions.

Substitute -2x + 7 for y in the first equation and solve the quadratic equation.

(x - 2)² + (-2x + 7 - 3)² = 20
(x - 2)² + (-2x + 4)² = 20
x² - 4x + 4 + 4x² - 16x + 16 = 20
5x² - 20x + 20 = 20
5x² - 20x = 0
5x(x - 4) = 0
x = 0 or x = 4

Find the matching y-values.

y = -2(0) + 7 = 7, (0,7)
y = -2(4) + 7 = -1, (4,-1)

36. On a certain tropical island, there are currently 500 palm trees and 200 flamingos. Suppose the palm tree population is decreasing at an annual rate of 3% per year and the flamingo population is growing at a continuous rate of 2% per year.

Write two functions, P(x) and F(x), that represent the number of palm trees and flamingos on this island, respectively, x years from now.

State the solution to the equation P(x) = F(x), rounded to the nearest year. Interpret the meaning of this value within the given context.

Answer:

Both functions are exponential. P can be written using the rate of decrease, compounded annually. Since F is continuous, e should be used.

P(x) = 500(1 - .03)^x
F(x) = 200e^.02x

For the second part, you can graph the two functions and find the intersection point. Since it needs to be to the nearest year, you can look at the table of values.

The two will intersect around (18.159, 287.580)

There will be the same number of flamingoes and palm trees in 18 years.

End of Part III

How did you do?

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now order 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.

June 2023 Algebra 2, Part II

This exam was adminstered in June 2023.

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Algebra 2 June 2023

Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


25. The business office of a local college wishes to determine the methods of payment that will be used by students when buying books at the beginning of a semester. Explain how the office can gather an appropriate sample that minimizes bias.

Answer:

The college should take a random sample survey from the list of students.

Your answer should imply sort of randomization and include an appropriate sample.

26. Determine the solution of √(3x + 7) = x - 1 algebraically.

Answer:

Square both sides of the equation to get rid of the radical and then solve the equation that results from the exponents being equal.

√(3x + 7) = x - 1
3x + 7 = x² - 2x + 1
0 = x² - 5x - 6
0 = (x - 6)(x + 1)
x = 6 or x = -1

Check both answers:

√(3(6) + 7) = √(25) = 5; 6 - 1 = 5, Check.
√(3(-1) + 7) = √(4) = 2; -1 - 1 = -2. discard as extraneous.

x = 6 is the only answer.



27. The population of bacteria, P(t), in hundreds, after t hours can be modeled by the function P(t) = 37e^0.0532t . Determine whether the population is increasing or decreasing over time. Explain your reasoning.

Answer:

You can graph this with your calculator and state that the graph is rising so the population is increasing. Or you can evaluate e^0.0532t.

Since e^0.0532t = 1.0546..., which is greater than 1, the population is increasing.

28. The polynomial function g(x) = x³ + ax² - 5x + 6 has a factor of (x - 3). Determine the value of a.

Answer:

Set the function equal to 0, substitute x = 3, and solve for a.

g(3) = (3)³ + a(3)² - 5(3) + 6 = 0
27 + a(9) - 15 + 6 = 0
9a = - 18
a = -2

29. Write a recursive formula for the sequence 189, 63, 21, 7, … .

Answer:

A recursive function needs and a₁ value followed by a_n defined in terms of a_n-1.

a₁ = 189
a_n = (1/3) a_n-1

30. Solve algebraically for x to the nearest thousandth:

2e^0.49x = 15

Answer:

Solve using inverse operations, including natural logs.

2e^0.49x = 15
e^0.49x = 15/2
ln(e^0.49x) = ln(15/2)
0.49x = 2.01490...
x = 4.112

31. For all values of x for which the expression is defined, write the expression below in simplest form.

(2x³ + x² - 18x - 9) / (3x - x²)

Answer:

Factor the numerator and the denominator and cancel out the common factors.

(2x³ + x² - 8x - 9) / (3x - x²)

( x²(2x + 1) - 9(2x + 1) ) / ( (x)(3 - x) )

( (x²- 9)(2x + 1) ) / ( (x)(3 - x) )

( (x + 3)(x - 3)(2x + 1) ) / ( (x)(-1)(x - 3) )

( (x + 3)(2x + 1) ) / ( (-x) )

32. An app design company believes that the proportion of high school students who have purchased apps on their smartphones in the past 3 months is 0.85. A simulation of 500 samples of 150 students was run based on this proportion and the results are shown below.

Suppose a sample of 150 students from your high school showed that 88% of students had purchased apps on their smartphones in the past 3 months. Based on the simulation, would the results from your high school give the app design company reason to believe their assumption is incorrect? Explain.

Answer:

Find the confidence interval by multiplying the standard deviation (SD) by 2. Subtract that from the mean to get the lower bound of the interval and add it to the mean to get the upper bound. See if 88% falls within that interval.

(2)(0.029) = 0.058

Lower bound = 0.852 - 0.058 = 0.794. Upper bound = 0.852 + 0.058 = 0.910.

Since 88% falls inside the confidence interval, the results would not give the company reason to believe that their assumption is incorrect.

End of Part II

How did you do?

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now order 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.

January 2023 Algebra 2 Regents Part IV

This exam was adminstered in January 2023.

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Algebra 2 January 2023

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 Objects cool at different rates based on the formula below.

T = (T₀ - T_R)e^2rt + T_R T₀: initial temperature T_R: room temperature r: rate of cooling of the object t: time in minutes that the object cools to a temperature, T
Mark makes T-shirts using a hot press to transfer designs to the shirts. He removes a shirt from a press that heats the shirt to 400°F. The rate of cooling for the shirt is 0.0735 and the room temperature is 75°F. Using this information, write an equation for the temperature of the shirt, T, after t minutes.
Use the equation to find the temperature of the shirt, to the nearest degree, after five minutes.
At the same time, Mark’s friend Jeanine removes a hoodie from a press that heats the hoodie to 450°F. After eight minutes, the hoodie measured 270°F. The room temperature is still 75°F. Determine the rate of cooling of the hoodie, to the nearest ten thousandth.
The T-shirt and hoodie were removed at the same time. Determine when the temperature will be the same, to the nearest minute.

Answer:

Write the equation substituting all values that we know. You should only have t remaining (and the letter e -- don't replace that with a number).

The equation will be T = (400 - 75)e^-.0735t + 75.

For the second part, substitute t = 5 and evaluate in your calculator:
T = (400 - 75)e^-.0735(5) + 75 = 300.05058... = 300 degrees

In the next part, you are given T and t but you need to find r:
270 = (450 - 75)e^-r(8) + 75
195 = (375)e^-8r
195/375 = e^-8r

log_e 195/375 = -8r
r = (log_e 195/375)/(-8)
r = 0.08174... = 0.0817.

For the last piece, we need to find t when the two expressions will be equal:
(375)e^-0.0817t + 75 = (325)e^-0.0735t + 75
(375)e^-0.0817t = (325)e^-0.0735t
e^-0.0817t = (325/375)e^-0.0735t
e^-0.0817t / e^-0.0735t = 13/15
e^-0.0082t = 13/15
ln 13/15 = -0.0082t
t = (ln 13/15) / (-0.0082) = 17.451...

About 17 minutes.

You could also have plugged each equation into your graphing calculator and compared the tables of values of the two equations. The intersection would happen at approximately 17 minutes. This will receive full credit if you explain where you got the answer from and likely allows fewer opportunites for mistakes.

As it is, I worked it out just to be sure that I could work it out. And in the end, I had an incorrect answer from putting the last step into the calculator incorrectly. Thankfully, I double-checked my work and discovered an error. Then I had to figure which of the two was incorrect!

End of Exam

How did you do?

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: 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.

School Life #39: Helium and Iron

(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?).

Learning was assessed, just not the learning that was expected.

This was just one of those things that pops up in Geometry class when things are written on the board, and they appear to be something else when looked at out of context. And someone who is less aware (or more asleep) might not have the proper context. (Reference "What's a Parabola?")

A new semester is under way. I hope it's fruitful. In the meantime, I'm catching up on old Regents exams so I can avoid rewriting something that needs to be rewritten even though I don't want to rewrite it. (Edit, sure, but rewrite?)

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 2023 Algebra 2 Regents, Part III

This exam was adminstered in January 2023.

More Regents problems.

Algebra 2 January 2023

Part III: Each correct answer will receive 4 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.


33. Solve the equation √(49 - 10x) + 5 = 2x algebraically.

Answer:

Subtract 5 from both sides to isolate the radical. Square both sides. Then solve the quadratic equation. Finally, discard extraneous solutions that were added when you squared the equation.

You do this by checking the answers that you got.

√(49 - 10x) + 5 = 2x

√(49 - 10x) = 2x - 5

49 - 10x = 4x² - 20x + 25

0 = 4x² - 10x - 24

0 = 2x² - 5x - 12

0 = (2x + 3)(x - 4)

2x + 3 = 0 or x - 4 = 0

x = -3/2 or x = 4

Check -3/2: √(49 - 10(-3/2)) + 5 ?= 2(-3/2)
√(49 - 10(-3/2) ?= -3 - 5 = -8
The square root cannot be negative, so reject this answer.

Check 42: √(49 - 10(4)) + 5 ?= 2(4)
√(9) + 5 ?= 8
3 + 5 = 8 (check!)
x = 4 is the only solution.

34. Joette is playing a carnival game. To win a prize, one has to correctly guess which of five equally sized regions a spinner will land on, as shown in the diagram below.

She complains that the game is unfair because her favorite number, 2, has only been spun once in ten times she played the game.

State the proportion of 2’s that were spun.

State the theoretical probability of spinning a 2.

The simulation output below shows the results of simulating ten spins of a fair spinner, repeated 100 times.

Does the output indicate that the carnival game was unfair? Explain your answer.

Answer:

Empirical probability is 1/10 because there was 1 positive result in 10 tries.

Theoretical probability is 1/5 because the number 2 covers one-fifth of the area of the wheel. There are five possible outcome and each should be as likely to occur as the others.

The chart shows that in 21 out of 100 simulations a result of 1/10, or 0.10 occured. This is not an unusual outcome.

If you think about it, with 5 possible outcomes, in 10 spins, a result of 2 should happen two times. If a result of 2 only happened one time, or even 3 times, in 10 spins that wouldn't be an unlikely occurrence because variations can be expected.

35. Graph c(x) = -9(3)^{x - 4} + 2 on the axes below.
Describe the end behavior of c(x) as x approaches positive infinity.
Describe the end behavior of c(x) as x approaches negative infinity.

Answer:

You can use your graphing calculator to get the table of values for where to plot the points. Your graph should look like this one below.

The end behavior of c(x): as x approaches positive infinity, c(x) approaches negative infinity. As x approaches negative infinity, c(x) approaches 2.

36. The monthly high temperature (°F) in Buffalo, New York can be modeled by B(m) = 24.9sin(0.5m - 2.05) + 55.25, where m is the number of the month and January = 1.
Find the average rate of change in the monthly high temperature between June and October, to the nearest hundredth.
Explain what this value represents in the given context.

Answer:

June is the 6th month and October is the 10th month. Find B(6) and B(10). Subtract B(10) - B(6) and divide the difference by (10 - 6), which is 4 to find the average monthly change.

B(6) = 24.9*sin(0.5(6) - 2.05) + 55.25 = 75.5040
B(10) = 24.9*sin(0.5(10) - 2.05) + 55.25 = 59.9915

(B(10) - B(6)) / 4 = (59.9915 - 75.5040) / 4 = -3.878 = -3.88

This means that for each month, the temperature is decreasing by 3.88 degrees.

End of Part III

How did you do?

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now order 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.

January 2023 Algebra 2, Part II

This exam was adminstered in January 2023.

More Regents problems.

Algebra 2 January 2023

Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


25. Algebraically determine the zeros of the function below.

r(x) = 3x³ + 12x² - 3x - 12

Answer:

Set the expression equal to 0. Then factor by grouping.

3x³ + 12x² - 3x - 12 = 0

3x³ - 3x + 12x² - 12 = 0

3x(x² - 1) + 12(x² - 1) = 0

(3x + 12)(x² - 1) = 0

(3x + 12)(x - 1)(x + 1) = 0

3x + 12 = 0 or x - 1 = 0 or x + 1 = 0

x = -4 or x = 1 or x = -1

26. Given a > 0, solve the equation a^{x + 1} = ∛(a²) for x algebraically

Answer:

Cube both sides of the equation to get rid of the radical and then solve the equation that results from the exponents being equal.

a^{x + 1} = ∛(a²)
a^{3x + 3} = a²
3x + 3 = 2
3x = -1
x = -1/3

27. Given P(A) = 1/3 and P(B) = 5/12, where A and B are independent events, determine P(A ∩ B).

Answer:

The probably of two independent events happening in the probability of one of them happening times the probability of the other.

P(A ∩ B) = (1/3)(5/12) = 5/36.

28. The scores on a collegiate mathematics readiness assessment are approximately normally distributed with a mean of 680 and a standard deviation of 120.

Determine the percentage of scores between 690 and 900, to the nearest percent.

Answer:

Use your graphing calculator. Find the function normalcdf.

Enter the following command: normalcdf(690,900,680,120) for the range minimum and maximum, followed by the median and the standard deviation. The answer will be 43%.

If you estimate it using a standard deviation chart, you won't get an exact answer.

29. Consider the data in the table below.

x	1	2	3	4	5	6
y	3.9	6	11	18.1	28	40.3

State an exponential regression equation to model these data, rounding all values to the nearest thousandth.

Answer:

Put the data into List 1 and List 2 on your graphing calculator. Run a Exponential Regression (ExpReg).

You should get the following output: a = 2.4585... and b = 1.6159...

The equation you want is y = (2.459)(1.616)^x

30. Write the expression A(x) • B(x) - 3C(x) as a polynomial in standard form.

A(x) = x³ + 2x - 1
B(x) = x² + 7
C(x) = x⁴ - 5x

Answer:

Multiply the first two expression A(x) and B(x). Subtract the product of 3 times C(x).

(x³ + 2x - 1)(x² + 7) - 3(x⁴ - 5x)
x⁵ + 7x³ + 2x³ + 14x - x² - 7 - 3x⁴ + 15x
x⁵ + 9x³ - x² + 14x - 7 - 3x⁴ + 15x
x⁵ - 3x⁴ + 9x³ - x² + 29x - 7

31. Over the set of integers, completely factor x⁴ - 5x² + 4

Answer:

The first step is to factor it the way you would factor y² - 5y + 4. Then factor the quadratic expression that result.

x⁴ - 5x² + 4

(x² - 4)(x² - 1)

(x - 2)(x + 2)(x - 1)(x + 1)

32. Natalia’s teacher has given her the following information about angle θ.

• π < θ < 2π • cos θ = &sqrt;(3)/4

Explain how Natalia can determine if the value of tan θ is positive or negative.

Answer:

Cosine is positive in Quadrants I and IV, and π < θ < 2π indicates Quadrants III and IV, so θ must put the angle into quadrant IV.

Tangent is negative in Quadrant IV.

End of Part II

How did you do?

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now order 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.

August 2023 Geometry Regents Part IV

This exam was adminstered in August 2023.

August 2023 Geometry, Part IV

A correct answer is worth 6 credits. Partial credit can be given for correct statements in the proof.

35. In the diagram below of quadrilateral FACT, BR intersects diagonal AT at E, AF || CT, and AF ≅ CT.

Prove: (AB)(TE) = (AE)(TR)

Answer:

This question is a little different, because most proofs of this type usuablly rely on find two traingles to be congruent via SSS, SAS, etc., and then showing two parts to be congruent using CPCTC. This one doesn't do that. This one asks you to prove a multiplication statement is true, but these products are the results of cross-multiplying the numerators and denominators of a proportion. You need to show that the two triangles are similar, not congruent, and then cross multiply the sides because the product of the means will equal the product of the extremes.

Also, notice that you are only told that this is a "quadrilateral", not a parallelogram. This was the case in June, as well. I wonder if it's going to be a recurring theme going forward.

Your proof should look like this:

Statement	Reasons
Quadrilateral FACT, BR intersects diagonal AT at E, AF || CT, and AF ≅ CT.	Given
FACT is a parallelogram	A quadrilateral with one pair of sides that are parallel and congruent is a parallelogram
∠ BAE ≅ ∠ RTE	Alternate Interior Angles
∠ AEB ≅ ∠ TER	Vertical Angles
△AEB ~ △RET	AA Postulate
(AB) / (AE) = (TR) / (TE)	Corresponding sides of similar triangles are proportional
(AB)(TE) = (AE)(TR)	The product of the means = the product of the extremes

Each statement is important. If you leave any out, you will lose one credit. However, if you have a couple of statements correct and the semblance of a proof, you should still earn two points for the question.

Also note that I wrote "The product of the means = the product of the extremes" instead of "cross multiplication", which is in essence the same thing, except that it is an instruction and not a justification or Reason (theorem, postualte, etc.). You could argue this point all you want, but there is an example in the Model Response Set where the only point lost is for the final step where "Cross multiply" is written, and the Regents literally calls out this step as an incorrect reason.

End of Exam

How did you do?

Questions, comments and corrections welcome.

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.

August 2023 Geometry Regents Part III

This exam was adminstered in August 2023.

August 2023 Geometry, Part III

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

32. Josh is making a square-based fire pit out of concrete for his backyard, as modeled by the right prism below. He plans to make the outside walls of the fire pit 3.5 feet on each side with a height of 1.5 feet. The concrete walls of the fire pit are going to be 9 inches thick.

If a bag of concrete mix will fill 0.6 ft³, determine and state the minimum number of bags needed to build the fire pit.

Answer:

First you need to find the Volume, which is length * width * height (rember that it has a sqware base) minus the volume of the hole in the center, which has the same height but is 18 inches less wide and long. Also keep in mind that they are mixing feet and inches in this problem. (Guess what! Real life does that, too!)

Nine inches is .75 feet, and 18 inches is 1.5 feet. Thus, 3.5 - 1.5 = 2.0 feet.

Volume = (3.5)(3.5)(1.5) - (2)(2)(1.5) = 12.375 ft³

If one bag of concrete mix fills 0.6 ft³, then the required number will be 12.375/0.6 = 20.625 or 21 when rounded up. You must round up or you will not have enough concrete mix to complete the job.

33. A telephone pole 11 meters tall needs to be stabilized with a support beam, as modeled below.

Two conditions for proper support are:

• The beam reaches the telephone pole at 70% of the telephone pole’s height above the ground.

• The beam forms a 65° angle with the ground.

Determine and state, to the nearest tenth of a meter, the length of the support beam that meets these conditions for this telephone pole.

Determine and state, to the nearest tenth of a meter, how far the support beam must be placed from the base of the pole to meet the conditions.

Answer:

First, find 70% of 11 meters (by multiplying) to find how far up the pole the beam will reach. Then use the sin 65° to find the length of the beam, and either tan 65° or the Pythagorean Theorem to find the distance on the ground.

11 * .7 = 7.7 meters.

sin 65° = 7.7 / x
x = 7.7 / sin 65° = 8.46 = 8.5 meters

tan 65° = 7.7 / y
y = 7.7 / tan 65° = 3.59 = 3.6 meters

Look at these answers. The beam is 8.5 meters, which is reasonable for the hypotenuse when the longer leg is 7.7. And the base is 3.6 meters, which is less than half of 7.7, which is reasonable for the side opposite a 25 degree angle. If the numbers you got seem extremely large or extremely small, you may have done one of three things: used radians instead of degrees, entered the ratio incorrectly (or used the wrong one), or forgot a decimal point somewhere.

If you got a negative answer, it's probably radians.

34. The coordinates of the vertices of quadrilateral ABCD are A(0,4), B(3,8), C(8,3), and D(5,-1).

Prove that ABCD is a parallelogram, but not a rectangle.
[The use of the set of axes below is optional.]

Answer:

Use the grid if you want to visualize the problem better. You can also use it to help with finding slopes.

A parallelogram has two pairs of parallel sides, and those sides will therefore have the same slope. In a rectangle, the sides must be perpendicular, which means that the slopes are inverse reciprocals and that the products of the slopes of the two lines will be -1.

Find all four slopes. Show that the opposite sides have the same slope. Show that the consecutive sides are not perpendicular. Then write your concluding statement.

Slope of AB = (8-4) / (3-0) = 4/3. Slope of BC = (3-8) / (8-3) = -5/5 = -1.
Slope of CD = (-1-3) / (5-8) = -4/-3 = 4/3. Slope of DA = (4-(-1)) / (0-5) = 5/-5 = -1.

AB || CD and BC || DA because they have the same slopes, therefore ABCD is a parallelogram.

Slope of AB * Slope of BC = (4/3)(-1) = -4/3. The slopes are not inverse reciprocals, so the lines are not perpendicular. Therefore, ABCD is NOT a rectangle.

Anther method:

Paralelograms have a lot of properties, as do rectangles. If you don't mind use the distance formula (Pythagorean Theorem, if you graphed ABCD), you can show that the opposites sides are congruent, which makes it a parallelogram, but the diagonals are NOT congruent, which means is could not be a rectangle.

This is a valid option. However, I'd prefer to use the slope formula over the distance formula is I could. You basically have to find the slope to use the distance formula.

End of Part III

How did you do?

Questions, comments and corrections welcome.

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.

August 2023 Geometry Regents Part II

This exam was adminstered in August 2023.

August 2023 Geometry, Part II

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

25. On the set of axes below, congruent quadrilaterals ROCK and R'O'C'K' are graphed.

Describe a sequence of transformations that would map quadrilateral ROCK onto quadrilateral R'O'C'K'.

Answer:

The figure has to move from Quadrant IV to Quadrant II. Inspection shows that the orientation of the quadrilateral has rotated. However, a rotation about the origin will not create the image of K at (-1,1), so a translation must follow.

Rotate ROCK 180 degrees around the origin. Then translate the image 2 to the left and 1 up.

There are many possible answers involding rotations around different centers, or reflections across the x-axis and y-axis or even other lines.

26. In triangle CEM, CE = 3x + 10, ME = 5x - 14, and CM = 2x - 6. Determine and state the value of x that would make △CEM an isosceles triangle with the vertex angle at E.

Answer:

If the vertex angle is E then sides CE and ME must be congruent and their lengths are equal.

Therefore,

5x - 14 = 3x + 10
2x = 24
x = 12

27. A flagpole casts a shadow on the ground 91 feet long, with a 53° angle of elevation from the end of the shadow to the top of the flagpole.

Determine and state, to the nearest tenth of a foot, the height of the flagpole.

Answer:

The height and the shadow mean that we need to use tangent. (We aren't concerned with the hypotenuse of the right triangle created.

The height is opposite the 53° angle and the shadow on the ground is adjacent to it.

tan 53 = x / 91
x = 91 tan 53 = 120.76... = 120.8 feet

28. A man is spray-painting the tops of 10 patio tables. Five tables have round tops, with diameters of 4 feet, and five tables have rectangular tops, with dimensions of 4 feet by 6 feet. A can of spray paint covers 25 square feet. How many cans of spray paint must be purchased to paint all of the tabletops?

Answer:

Find the area of one circle (πr²) and multiply by 5. Find the area of one rectangle (A=LW) and multiply by 5.

A = 5π(2)² + 5(4)(6) = 182.83

Divide this by 25 and round up to the next can: 182.83 / 25 = 7.3...

8 cans of paint are needed.

If you don't round up, you will not have enough paint to finish the job.

29. Using a compass and straightedge, construct a midsegment of △AHL below. [Leave all construction marks.]

Answer:
To construct a midsegment, you need two midpoints. To get two midpoints, you need to construct two perpendicular bisectors. You have a choice of which side you want to bisect. You may NOT use a ruler to measure the midpoints.

From point H, make an arc that is more than half the length of HL and swing it across HL and HA. (You could make two separate arcs, but there's no reason not to use the same one.

From point A, make an arc that intersects the first arc in two places. From point L, construct a third arc that intersect the arc drawm from point H.

Draw the two perpendicular bisectors. These will give you the midpoints. Use a straightedge to connect the midpoints. This is the midsegment, which is half the size of AL and parallel to AL.

30. Right triangle STR is shown below, with m∠T = 90°. Altitude TQ is drawn to SQR, and TQ = 8.

If the ratio SQ:QR is 1:4, determine and state the length of SR.

Answer:
Label SQ x and QR 4x. The product of (SQ)(QR) is equal to the square of (QT) by the Right Triangle Altitude Theorem.

(x)(4x) = 8²

4x² = 64

x² = 16

x = 4

SR is x + 4x = 5x, and x = 4, so 5x = 5(4) = 20. SR = 20.

31. Line AB is dilated by a scale factor of 2 centered at point A.

Evan thinks that the dilation of AB will result in a line parallel to AB, not passing through points A or B.

Nathan thinks that the dilation of AB will result in the same line, AB.

Who is correct?

Explain why.

Answer:
Nathan is correct because every point on line AB will move twice as far from A in the same direction along the line AB. Also, since the image of point A will conincide with point A, parallel lines aren't possible.cimal because it's infinite.

End of Part II

How did you do?

Questions, comments and corrections welcome.

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.