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

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Algebra Regents

Part I

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


1.What is the correct factorization of x² + 4x - 12?

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

Answer: (4) (x - 2)(x + 6)

What two factors of -12 have a sum of +4? One factor must be negative and the other must be positive, and the "larger" number must be positive.

Choice (1) is x² - x - 12.

Choice (2) is x² + x - 12.

Choice (3) is x² - 4x - 12.

Choice (2) is x² + 4x - 12. This is the correct answer.

2. Which situation can be modeled by a linear function?

(1) A printer can print one page every three seconds.
(2) A bank account earns 0.5% interest each year, compounded annually.
(3) The number of cells in an organism doubles every four days.
(4) The attendance at a professional sports team’s games decreases by 1.5% each year.

Answer: (1) A printer can print one page every three seconds.

The linear function never changes. It has a constant rate. One page every three seconds is a constant rate of change for the total number of pages printed. The correct answer is Choice (1).

Choice (2) is not linear because the interest is compounded, meaning interest is earned on the interest.

Choice (3) is not linear because the amount doubles every day, which is not a constant rate of range.

Choice (4) is not linear because the decrease is compounded from year to year, losing 1.5% of the attendance of the year before.

3. Which expression is equivalent to 3(x² - 2x + 3) - (4x² + 3x - 1)?

(1) -x² + x + 2
(2) -x² - 8x + 7
(3) -x² - 3x + 8
(4) -x² - 9x + 10

Answer: (4) -x² - 9x + 10

Distribute the 3 and the minus sign and then combine like terms.

3(x² - 2x + 3) - (4x² + 3x - 1)
3x² - 6x + 9 - 4x² - 3x + 1
-x² - 9x + 10

This is Choice (4).

4. At Adelynn’s first birthday party, each guest brought $1 in coins for her piggy bank. Guests brought nickels, dimes, and quarters for a total of $28. There were twice as many dimes as nickels and 12 more quarters than nickels. Which equation could be used to determine the number of nickels, x, that her guests brought to her party?

(1) .05x + .10x + .25x = 28
(2) .05x + .10(2x) + .25(x + 12) = 28
(3) .05(2x) + .10x + .25(x + 12) = 28
(4) .05(x + 12) + .10(2x) + .25x = 28

Answer: (2) .05x + .10(2x) + .25(x + 12) = 28

If x represents the number of nickels, then the value of the nickels is .05x. Eliminate Choices (3) and (4).

Choices (1) states that there are the same number of nickels, dimes, and quarters, which is not true. Eliminate Choice (1).

Choice (2) is the correct answer because there are 2 times as many dimes as nickels and 12 more quarters.

5. A student creates a fourth-degree trinomial with a leading coefficient of 2 and a constant value of 5. The trinomial could be

(1) 2x⁴ + 3x² + 5
(2) 2x⁴ + 5x + 3
(3) 4x² - 3x + 5
(4) 4x³ - 5x² + 3

Answer: (1) 2x⁴ + 3x² + 5

Fourth-degree means that the polynomial has x⁴ as the term with the largest exponent. Leading coefficient of 2 means that it will have 2x⁴. The constant has no variable and has to equal 5.

The only possibility is Choice (1).

Choice (2) has a constant of 3, not 5.

Choices (3) and (4) do not have a leading coefficient of 2, nor are they fourth-degree.

6. When solving the equation 4x² - 16 = 0, Laura wrote 4x² = 16 as her first step. Which property justifies Laura’s first step?

(1) distributive property of multiplication over addition
(2) multiplication property of equality
(3) commutative property of addition
(4) addition property of equality

Answer: (4) addition property of equality

Whether or not this is how you would solve this equation, the question is about Laura solving the problem. The first thing she did was add 16 to both sides of the equation. She can do this because of the Addition Property of Equality.

The correct answer is Choice (4).

7. Which expression results in an irrational number?

Answer: (4) 1/3 + √(3)

The sum of a rational and an irrational number is always irrational.

Choice (1) has a value of 3, which is rational.

Choice (2) has a value of 45, which is rational.

Choice (3) has a value of -5/12, which is rational.

8. Which equation has the same solutions as x² + 6x - 18 = 0?

(1) (x + 3)² = 24
(2) (x + 3)² = 27
(3) (x + 6)² = 24
(4) (x + 6)² = 27

Answer: (2) (x + 3)² = 27

Complete the Square:

x² + 6x - 18 = 0
x² + 6x = 18
x² + 6x + 9 = 27
(x + 3)² = 27

Half of 6 is 3 and 3 squared is 9. Also, x² + 6x + 9 is a perfect square, namely (x + 3)².

The correct choice is Choice (2).

More to come. Comments and questions welcome.

Writing Update I will be at Philcon this weekend, November 23-34, 2024, in Cherry Hill, NJ, at the DoubleTree by Hilton. I'll be at TWO book launch parties: one that will include my book (see the table below), and another for an anthology that I have a story in. If you're in the area, come on by!

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!

School Life #43: Two Wrong

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

Did he say he got two wrong ... or that he was just too wrong.

Proprtionality is important. Percentages are important.

Only getting 2 wrong out of 25 isn't a big dead (except to the highly competitive, and I totally respect that). Getting 2 wrong out of 3 or 4 means you need to do more reviewing.

Yes, a regular schedule of updates to this site would be nice. I'm not sure when that will happen. Working on it.

Writing Update I will be at Philcon this weekend, November 23-34, 2024, in Cherry Hill, NJ, at the DoubleTree by Hilton. I'll be at TWO book launch parties: one that will include my book (see the table below), and another for an anthology that I have a story in. If you're in the area, come on by!

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!

Come back often for more funny math and geeky comics.

Social Media Logic Leap

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

"And you know what they're really saying..."

It's not just social media, but it's bad there. You take one sentence or a two-second clip, and then you stretch it, distort it and refute the distortions.

I've seen a two-second clip with commentary like, "Really? Does that mean that X? Won't that lead to Y and Z?" And they're on a roll. All this despite the fact that if you look at a seven-second clip or - Heaven Forbid! - 30 seconds or the complete response, it's quite obvious that, no, it doesn't mean X, Y, or even Z, whatever those stand for.

And even if those points aren't addressed, no one will ever ask. They'll just accuse because that's easier.

That's why I do math and apply actual logic. I prefer the challenge.

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!

Come back often for more funny math and geeky comics.

You're Back!

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

"For how long, Mr. Spock? ... For how long?"

I've made promises, and I've broken them. But since I don't know how many people are still reading blogs these days, I can't tell how many people, if any, that I've disappointed. Few have inquired about my extended absences, so it leaves me wondering.

Two things spurred me on: I really want to make it to 2,000 comics, which I should have surpassed a long time ago, and I got an email from Fables dot pro about hosting my comic. Fables got my name from the old TopWebComics site, which still exists, actually, even if I haven't visited it in over a decade. (By contrast, I still check out the Comic Genesis site a few times per year, even if I haven't updated it. And I'm not even sure I remember how to update it. I might need my old PC where there's a saved password.)

More to come. For real this time.

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!

Come back often for more funny math and geeky comics.

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

This exam was adminstered in August 2024 .

August 2024 Algebra 2, Part IV

A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

37. Taylor wants to open an investment account with the $1200 she received for her birthday. She has narrowed her choices down to two banks. America’s Bank offers 6.4% annual interest compounded quarterly. Barnyard Bank offers 6.35% annual interest compounded continuously.

Write functions for A(t) and B(t) to represent the value of her investment with America’s Bank and Barnyard Bank as a function of time, t, in years.

Taylor would like to invest the $1200 into one bank for ten years making no additional deposits and no withdrawals. With which bank will Taylor earn the most money? Justify your answer.

Taylor chooses to invest her money in Barnyard Bank. Algebraically determine how long, to the nearest tenth of a year, it will take her initial investment to triple assuming she makes no deposits or withdrawals.
Justify your answer.

Answer:

Compounded quarterly means four times a year. Compounded continuously means you have to use e.

A(t) = 1200(1 + 0.064/4)^4t
B(t) = 1200e^0.0635t

A(10) = 1200(1 + 0.064/4)⁴⁽¹⁰⁾ = 2264.277...
B(t) = 1200e^0.0635(10) = 2264.426...

After 10 years, Barnyard Bank will earn more money.

Triple $1200 is $3600. Some for B(t) = 3600.

3600 = 1200e^0.0635t
3 = e^0.0635t
ln 3 = 0.0635t
ln 3 / 0.0635 = t

t = 17.301...

It will take 17.3 years for the investment to triple.

End of Exam

How did you do?

Questions, comments and corrections 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 III)

This exam was adminstered in August 2024 .

August 2024 Algebra 2, Part III

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

33. The population of China, in millions, can be modeled by the function P(x) = 316.93e^0.0133x , where x is the number of years since 1900. The population of India since 1900 is summarized in the table below:

Which country’s population had a greater average rate of change between 1950 and 2020? Justify your answer.

Answer:

Find the rate of change for India by subtracting the population in 1950 from the population in 2020, and divide it by 70 (years). Find the change for China by evaluating the function for x = 50 and x = 120 (years since 1900) and divide that by 70. Which is bigger?

Note that you can solve the problem without dividing each number by 70, but they specifically asked about the rates, so divide by 70.

India: (1380 - 376.3) / 70 = 1003.7 / 70

China: P(50) = 316.93e^0.0133(50) = 616.267
P(120) = 316.93e^0.0133(120) = 1563.498
(P(120) - P(50)) / 70 = (1563.498 - 616.267) / 70 = 947.231 / 7

India had a greater rate of change because 1003.7 / 70 > 947.231 / 70.

34. In a packaging plant, a machine packs boxes with jars. The machine’s manufacturer states that a box is packed, on average, every 42 seconds. To test that claim, the packaging plant randomly selects a sample of 10 boxes and finds the sample mean to be 49.8 seconds.

The company ran a simulation of 1000 trials based on the manufacturer's claim. The approximately normal results are shown below.

Based on the simulation, determine an interval containing the middle 95% of plausible mean times. Round your answer to the nearest hundredth.

Is the time 49.8 seconds unusual? Use statistical evidence to justify your answer.

Answer:

The middle 95% is the mean minus 2 times the standard deviation to the mean plus 2 times the standard deviation.

42.029 - 6.21 = 35.819, 42.029 + 6.21 = 48.239. (35.82, 48.24) to the nearest hundredth.

The value 49.8 is unusual because it falls outside the middle 95%. (It is greater than 48.24.)

35. Consider the function f(x) = 2^x.
Is f(x) an even function? Justify your answer.

Write an equation for g(x), the function that results after f(x) is shifted up 5 units.

Write an equation for h(x), the inverse of g(x).

Answer:

The function f(x) is not an even function because f(x) =/= f(-x). For example f(1) = 2, f(-1) = 1/2.

g(x) = f(x) + 5 = 2^x + 5

Note: I checked the official response guide and "f(x) + 5" was acceptable as an answer.

To find the inverse, replace x with y, and set the function equal to y. Then solve for x.

x = 2^y + 5
x - 5 = 2^y
log₂ (x - 5) = y

h(x) = log₂ (x - 5)

36. Solve the system of equations shown below algebraically:

(x - 4)² + (y - 1)² = 9
x - y = 6

Answer:

Find the two points on the line that intersect the circle.

Solve for x and substitute in the circle equation.

x - y = 6
x = y + 6

(x - 4)² + (y - 1)² = 9
(y + 6 - 4)² + (y - 1)² = 9
(y + 2)² + (y - 1)² = 9
y² + 4y + 4 + y² - 2y + 1 = 9
2y² + 2y + 5 = 9
2y² + 2y - 4 = 0
y² + y - 2 = 0
(y + 2)(y - 1) = 0
y + 2 = 0 or y - 1 = 0
y = -2 or y = 1

If y = -2, x - (-2) = 6, then x = 4. (4, -2)

If y = 1, x - 1 = 6, then x = 7. (7, 1)

End of Part III

How did you do?

Questions, comments and corrections 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 II)

This exam was adminstered in August 2024 .

August 2024 Algebra 2, 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 axes below, graph y = 3.2(1.8)² .

Answer:

Put the equation in your graphing calculator and check the Table of Values.

Points on the graph include: (0,3.2), (1,5.76), (2,10.368), (3,18.6624), (-1, 1.78), (-2,.99), (-3,.55), (-4,.3), (-5,.16). Approxminate these points.

26. Is x + 3 a factor of 7x³ + 27x² + 9x - 27?
Justify your answer.

Answer:

You can show if it is a factor or not by doing doing polynomial division. However, an easier method is this: if (x + 3) is a factor, then the polynomial will equal zero when x = -3.

7(-3)³ + 27(-3)² + 9(-3) - 27 = 0, so (x + 3) is a factor of the polynomial.

27. Over the set of integers, factor the expression 2x⁴ - 10x³ + 3x² - 15x completely.

Answer:

Factor completely means that there will be multiple steps. Over the set of integers means that you don't have to worry about imaginary numbers.

2x⁴ - 10x³ + 3x² - 15x
x(2x³ - 10x² + 3x - 15)
x(2x²(x - 5) + 3(x - 5))
x(2x² + 3)(x - 5)

28. The monthly unemployment rate of towns in the United States is approximately normally distributed with a mean rate of 5.2% and a standard deviation of 1.6%. Determine the percentage of towns, to the nearest integer, that have a monthly unemployment rate greater than 6%.

Answer:

Use the "normalcdf" function on your graphing calculator.

Use the command normalcdf(6,100,5.2,1.6) to find the percentage between 6% and 100% ("greater than 6%), mean 5.2%, standard deviation 1.6%.

The result should be .3085, which is .31, or 31%.

29. The function d(t) = 2cos(π/6 t) + 5 models the water depth, in feet, at a location in a bay, t hours since the last high tide. Determine the minimum water depth of the location, in feet, and justify your answer.

Answer:
The midline is 5 feet (from the +5). The amplitude is 2 feet (from 2cos). This means that the water depth varies between 3 feet and 7 feet.

Therefore, the minimum water depth is 3 feet.

Remember to justify it. This is easy enough to do in your head. Do NOT just write 3 feet, or 5 - 2 = 3. Indicate where the 5 and 2 came from and what they stand for.

30. A brewed cup of coffee contains 130 mg of caffeine. The half-life of caffeine in the bloodstream is 5.5 hours. Write a function, C(t) to represent the amount of caffeine in the bloodstream t hours after drinking one cup of coffee

Answer:
The function is expoonential, with 130 as the initial amount and 1/2 as the rate (half-life). It takes 5.5 hours for 1/2 to leave the bloodstream, but t is measured in hours, so we need to divide t by 5.5.

C(t) = 130(1/2)^t/5.5

31. Markus is a long-distance walker. In one race, he walked 55 miles in t hours and in another race walked 65 miles in t + 3 hours. His rates are shown in the equations below.

r = 55/t ; r = 65 / (t + 3)

Markus walked at an equivalent rate, r, for each race. Determine the number of hours that each of the two races took.

Answer:
Write a proportion and solve it.

55/t = 65/(t + 3)
65t = 55(t + 3)
65t = 55t + 165
10t = 165
t = 16.5

The first race took 16.5 hours, and the second race took 19.5 hours.

32. Solve the equation x² + 3x + 11 = 0 algebraically. Express the answer in a + bi form.

Answer:
Use the Quadratic Formula to solve this.

x = (-b + √(b² - 4ac) / (2a)
x = (-3 + √((3)² - 4(1)(11)) / (2(1))
x = (-3 + √(9 - 44)) / (2)
x = (-3 + √(-35)) / (2)
x = -3/2 + √(35)/2 i

Note that you MUST separate the expression into two fraction because the question states a + bi form.

End of Part II

How did you do?

Questions, comments and corrections 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!