## Sunday, July 30, 2023

### Algebra Problems of the Day (Algebra 1 Regents, June 2023 Part I)

The following questions appeared on the une 2023 Algebra 1 Regents Exam

More Regents problems.

### Algebra 1 Regents, June 2023

17. Some adults were surveyed to find out if they would prefer to buy a sports utility vehicle (SUV) or a sports car. The results of the survey are summarized in the table below.

Of the number of adults that preferred sports cars, approximately what percent were males?

1) 15.8
2) 45.2
3) 64.4
4) 82.6

84 adults preferred sports cars, and 38 of them were male, which is less than half, so eliminate Choices (3) and (4).

Divide 38/84 = 0.452..., which is about 45.2%, which is Choice (2).

18. The solution to 2x2 = 72 is

1) {9,4}
2) {-4. 9}
3) {6}
4) {+6}

You could test 4, 6 and 9 and see which numbers work.

Or you can solve it as follows:

2x2 = 72
x2 = 36
x = +6.

The correct answer is Choice (4).

While (4)(9) = 36, neither 42 nor 92 equal 36.

It's important to remember the negative solution. Positive 6 isn't the ONLY solution.

19. Three quadratic functions are given below.

Which of these functions have the same vertex?

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

Answer: 3) I and III, only

The vertex for function III can be seen to be (-2,5). Function I is written in vertex form, and the vertex for I is (-2,5). So I and III are the same. Eliminate Choices (1) and (2).

Next, determine if the vertex for function II is also (-2,5). You can see in the table that (-2,5) is a point in the function. HOWEVER, it is NOT the vertex. There is another point at (-1,5). That means that the Axis of Symmetry for function II occurs at x = -1.5. And the vertex of the function is on the Axis of Symmetry. Eliminate Choice (4).

The correct answer is Choice (3).

20. The domain of the function f(x) = x2 + x - 12 is

1) (-∞,-4]
2) (-∞,∞)
3) {-4,3]
4) [3,∞)

The domain is the set of all x values that are valid for the function. In this example, every value of x is valid, from negative infinity to positive infinity, which is Choice (2).

If the funtion contained, for example, a variable in the denominator of a fraction, or a variable under a square root symbol, then there would be values of x for which the function would not make sense and they would be outside of the function's domain.

The function has ZEROES at x = -4 and x = 3, but that is not the domain.

Choices (1) and (4) are portions of the function where the range (the y value) is non-negative. But these have nothing to do with the domain.

21. A father makes a deal with his son regarding his weekly allowance. The first year, he agrees to pay his son a weekly allowance of \$10. Every subsequent year, the allowance is recalculated by doubling the previous year’s weekly allowance and then subtracting 8. Which recursive formula could be used to calculate the son’s weekly allowance in future years?

1) an = 2n - 8
2) an = 2(n + 1) - 8
3) a1 = 10, an+1 = 2an - 8
4) a1 = 10, an+1 = 2(an - 8)

Answer: 3) a1 = 10, an+1 = 2an - 8

A recursive formula needs a beginning, which is a1. Each value after that is calculated from the one before it. Eliminate Choice (1) and (2).

The difference between Choices (3) and (4) is that in (3) the value is doubled and then 8 is subtracted. In (4), 8 is subtracted first and then the amount is doubled.

The definition of the formula agrees with Choice (3), which is the correct answer.

22. AWhat is the solution to the inequality below?

4 - 2/5 x > 1/3 x + 15

1) x < 11
2) x > 11
3) x < -15
4) x > -15

You could put each half of the equation into your graphing calculator to find where they are equal and see where the left side is greater than the right side.

4 - 2/5 x > 1/3 x + 15
-2/5 x > 1/3 x + 11
-2/5 x - 1/3 x > 11
-6/15 x - 5/15 x > 11
-11/15 x > 11
1x < (-15/11)11
x < -15

Choice (3) is the correct answer.

Remember to multiply by the reciprocal of the fraction. Remember that when you multiply an inequality by a fraction, you have to flip the inequality symbol the other direction.

23. Which statement is correct about the polynomial 3x2 + 5x -2?

1) It is a third-degree polynomial with a constant term of 22.
2) It is a third-degree polynomial with a leading coefficient of 3.
3) It is a second-degree polynomial with a constant term of 2.
4) It is a second-degree polynomial with a leading coefficient of 3.

Answer: 4) It is a second-degree polynomial with a leading coefficient of 3.

The degree of the polynomial has to do with the highest exponent in the expression, not the number of terms it contains. The leading coefficient is the coefficient of the term with the highest exponent. The constant is the term without a variable and the sign in front of it is part of the constant.

The polynomial 3x2 + 5x -2 is second degree with a leading coefficient of 3 and a constant of -2. The only choice that does not contradict any of this is Choice (4).

24. A store manager is trying to determine if they should continue to sell a particular brand of nails. To model their profit, they use the function p(n), where n is the number of boxes of these nails sold in a day. A reasonable domain for this function would be

1) nonnegative integers
2) rational numbers
3) real numbers
4) integers

The store manager cannot sell fractions of boxes of nails and he can't sell negative boxes of nails. Irrational boxes is right out.

He can sell zero boxes.

Nonnegative integers is the reasonable domain for this function. The correct answer is choice (1).

End of Part I. How did you do?

More to come. Comments and questions welcome.

More Regents problems.

### I also write Fiction!

You can now order Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
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.  ## Saturday, July 29, 2023

### Algebra Problems of the Day (Algebra 1 Regents, June 2023 Part I)

The following questions appeared on the une 2023 Algebra 1 Regents Exam

More Regents problems.

### Algebra 1 Regents, June 2023

9. A company ships an average of 30,000 items each week. The approximate number of items shipped each minute is calculated using the conversion

To get items per minute, you need to have items in the numerator and minutes in the denominator. Every other unit needs to cancel out (reduce to 1) by appearing once in each of the numerator and the denominator. You can cancel units like factors.

Choice (1) has minutes in the numerator, and two weeks in the denominator, which would give you "weeks2", which is meaningless. Eliminate Choice (1).

Choice (2) has items in the numerator and minutes in the denominator. It also has weeks, days, and hours in both the numerator and the denominator, so they they cancel out. Choice (2) is the correct answer.

Choices (3) and (4) have items in the denominator. Eliminate Choices (3) and (4).

10. A function is graphed below.

A possible equation for this function is

1) f(x) = (x + 2)(x - 3)
2) f(x) = (x - 2)(x + 3)
3) f(x) = (x - 2)2(x + 3)
4) f(x) = (x - 2)(x + 3)(x - 12)

3) f(x) = (x - 2)2(x + 3)

The zeroes of the function are at x = -3 and x = 2, which makes (x + 3) and (x - 2) factors. This is a cubic function, so you no that there is a third factor. (In reality, those fact checking me will tell you that it's obviously a function with an odd number exponent greater than 1, not an even number exponent. We aren't dealing with those graphs in Algebra 1, so don't worry about that right now.)

Basically, it's not a parabola, so it's not (1) or (2).

Choice (4) uses the y-intercept for no particular reason except to confuse you. There is no zero at x = 12. The graph indicates that it continues upward.

The fact that there is a minimum point is (2,0) is the reason that there are two factors of (x - 2), which is (x - 2)2. If you multiply (-2)(-2)(3), you will get +12, which is the expected y-intercept on the graph.

Choice (3) is the correct answer.

As with similar problems, you could put all four choices into your graphing calculator and see which one looks like the given graph.

11. If g(x) = -x2 - x + 5, then g(-4) is equal to?

1) -15
2) -7
3) 17
4) 25

Put the function in your calculator and look at the table of values for x = -4. Or just enter (-4)2 - (-4) + 5. Make sure that you use parentheses.

(-4)2 - (-4) + 5 = 16 + 4 + 5 = 25, which is Choice (4).

12. A movie theater’s popcorn box is a rectangular prism with a base that measures 6 inches by 4 inches and has a height of 8 inches. To create a larger box, both the length and the width will be increased by x inches. The height will remain the same. Which function represents the volume, V(x), of the larger box?

1) V(x) = (6 + x)(4 + x)(8 + x)
2) V(x) = (6 + x)(4 + x)(8)
3) V(x) = (6 + x) + (4 + x) + (8 + x)
4) V(x) = (6 + x) + (4 + x) + (8)

Answer: 2) V(x) = (6 + x)(4 + x)(8)

Volume equals length times width times height, so eliminate Choice (3) and (4).

The height of the box doesn't change, so there is no reason to add x to the 8 inches of the original box. Eliminate Choice (1).

Choice (2) show the length and the width increased by the same x inches while the height remains its orignal 8 inches. Choice (2) is the correct answer.

13. The expression 300(4)x + 3 is equivalent to

1) 300(4)x(4)3
2) 300(4x)3
3) 300(4x) + 300(43)
4) 300x(43)

If you mutliply two terms with the same base, you add the exponents. So multiplying (4)x(4)3 will give you (4)x + 3. Choice (1) is the correct answer.

If you don't believe that, you can try it on your calculator using any number in place of x, or enter both expressions as graphs and look at the matching tables of values.

Choice (2) can be rewritten as 300(4)3x, which isn't what we are looking for.

Choice (3) can be rewritten as 300(4x + 43), which isn't what we want.

Choice (4) is just nutty. It's there for people who don't evenread the questions before they choose an answer. Why would 300 suddenly gain an exponent of x?

14. Ashley only has 7 quarters and some dimes in her purse. She needs at least \$3.00 to pay for lunch. Which inequality could be used to determine the number of dimes, d, she needs in her purse to be able to pay for lunch?

1) 1.75 + d > 3.00
2) 1.75 + 0.10d > 3.00
3) 1.75 + d < 3.00
4) 1.75 + 0.10d < 3.00

Answer: 2) 1.75 + 0.10d > 3.00

"At least" means that you want to have an amount that is greater than or equal to \$3.00. Eliminate Choice (3) and (4).

Ashley doesn't want to add the number of dimes to the \$1.75. She needs to add the value of those dimes, which is \$0.10 times the number of dimes, d, or 0.10d.

Choice (2) is the correct answer.

15. The formula for the area of a trapezoid is A = 1/2(b1 + b2)h.
The height, h, of the trapezoid may be expressed as

You can solve a literal equation in the same manner that you solve an equation with one variable. The only difference is that you most likely won't combine any "like terms".

To solve for h, isolate the variable by using inverse operations.

A = 1/2(b1 + b2)h

2A = (b1 + b2)h

2A / (b1 + b2) = h

This is Choice (4).

16. The function f(x) = |x| is multiplied by k to create the new function g(x) = k|x|. Which statement is true about the graphs of f(x) and g(x) if k = 1/2?

1) g(x) is a reflection of f(x) over the y-axis
2) g(x) is a reflection of f(x) over the x-axis.
3) g(x) is wider than f(x).
4) g(x) is narrower than f(x).

Answer: 3) g(x) is wider than f(x).

As with a parabola, if the coefficient is between 0 and 1, the function will be wider than f(x) = |x|. This is Choice (3).

If k = -1, then g(x) would be a reflection of f(x) over the x-axis.

A reflection of f(x) = |x| over the y-axis would be equal to f(x) itself.

If k > 1 (or k < -1) then g(x) would be narrower than f(x).

More to come. Comments and questions welcome.

More Regents problems.

### I also write Fiction!

You can now order Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
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.  ## Thursday, July 27, 2023

### Algebra Problems of the Day (Algebra 1 Regents, June 2023 Part I)

The following questions appeared on the une 2023 Algebra 1 Regents Exam

More Regents problems.

### Algebra 1 Regents, June 2023

1. The expression 9m2 - 100 is equivalent to

1) (3m - 10)(3m + 10)
2) (3m - 10)(3m - 10)
3) (3m - 50)(3m + 50)
4) (3m - 50)(3m + 50)

Answer: 1) (3m - 10)(3m + 10)

The given expression is a difference of two perfect squares, which factors into two conjugates, two expressions with the same terms, but one has a plus and the other has a minus.

If you look at the choices, only one has a pair of constants that will have a product of -100 when multiplied, and that is Choice (1).

Choice (2) has a product of 9m2 - 60m + 100.
Choice (3) has a product of 9m2 - 2500.
Choice (4) has a product of 9m2 - 300m + 2500.

2. Which expression represents an irrational number?

1) √(16) + √(1)
2) √(25) + √(4)
3) √(36) + √(7)
4) √(49) + √(9)

The sum of two rational numbers is always rational. The sum of a rational and in irrational number is always irrational.

If you take the square root of a perfect square, you will get a rational number. Out of all the numbers which appear in the choices above, only √(7) is irrational. So the correct answer is Choice (3).

3. Which linear equation represents a line that passes through the point (-3,-8)?

1) y = 2x - 2
2) y = 2x - 8
3) y = 2x + 13
4) y = 2x - 14

Answer: 1) y = 2x - 2

You could put all four equations into your graphing calculator and check the Table of Values for the correct equation for x = -3. Or you can just substitute x = -3 into each of this equations. Since they all have the same slope, at most one of the equations will work.

y = 2(-3) - 2 = -8. This is correct, so the answer is Choice (1).
y = 2(-3) - 8 = -14, so (-3,-8) is not on this line.
y = 2(-3) + 13 = 7, so (-3,-8) is not on this line.
y = 2(-3) - 14 = -20, so (-3,-8) is not on this line.

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

1) 2x2 - 2x + 2
2) 2x2 + 2x + 10
3) 2x4 - 2x2 + 2
4) 2x4 - 2x2 + 10

Answer: 2) 2x2 + 2x + 10

When subtracting polynomials, the exponents DO NOT change. Eliminate Choices (3) and (4) immediately.

When you use the Distributive Property on the second expression, make sure you carry the minus sign. You are multiplying by NEGATIVE three.

(5x2 - x + 4) - 3(x2 - x - 2)
5x2 - x + 4 - 3x2 + 3x + 6
2x2 + 2x + 10

Choice (2) is the correct answer.

5. 5 The 24th term of the sequence 25, 211, 217, 223, … is

1) -149
2) -143
3) 133
4) 139

Each given term is negative and the value is decreasing. So the 24th term will be a negative number. Eliminate Choices (3) and (4).

Each term in the sequence is 6 less than the one before it. The common difference is -6.

If the first term is -5, and the common difference is -6, then the 24th term will be -5 - 6(23), because you have to subtract -6 twenty-three times to get to the 24th term.

-5 - 6(23) = -143, which is Choice (2).

If you made a mistake and multiplied by 24, you would've gotten -149, which is Choice (1), which would be an incorrect response.

6. When completing the square for x2 - 18x + 77 = 0, which equation is a correct step in this process?

1) (x - 9)2 = 4
2) (x - 3)2 = 2
3) x = +13
4) (x - 9) = +9

Answer: 1) (x - 9)2 = 4

When you Complete the Square, you need half of the coefficient of the middle term (when the x2 term has a coefficient of 1). So (x - 3)2 is incorrect, and can be eliminated.

To Complete the Square, do the following:

x2 - 18x + 77 = 0
x2 - 18x = -77
(half of -18 is -9, and (-9)2 = +81)

x2 - 18x + 81 = -77 + 81
x2 - 18x + 81 = 4
(x - 9)2 = 4

This is Choice (1).

If you expand (x - 9)2 = 4, you get
x2 - 18x + 81 = 4, which when you subtract 4 from both sides becomes,
x2 - 18x + 77 = 0.

7. Which function will have the greatest value when x > 1?

1) g(x) = 2(5)x
2) f(x) = 2x + 5
3) h(x) = 2x2 + 5
4) k(x) = 2x3 + 5

You could put all four equations into your graphing calculator and look at the results on the screen and in the table of values.

It should be obvious that quadratic functions will rise faster than linear functions when x > 1. Also, cubic functions will rise faster than quadratic functions when x > 1. Finally, exponential functions will rise faster than cubic functions when x > 1.

The correct answer is Choice (1).

When x = 1, f(1) = h(1) = k(1) = 7, and g(1) = 10.

8. Mike uses the equation b = 1300(2.65)x to determine the growth of bacteria in a laboratory setting. The exponent represents

1) the total number of bacteria currently present
2) the percent at which the bacteria are growing
3) the initial amount of bacteria
4) the number of time periods

Answer: 4) the number of time periods

The exponent is how many times you are going to multiply by 2.65.

The total number of bacteria is b.

The percent at which the bacteria are growing is 165%. (That is, 1 + 1.65 = 2.65.)

The initial amount of bacteria is 1300.

More to come. Comments and questions welcome.

More Regents problems.

### I also write Fiction!

You can now order Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
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.  ## Wednesday, July 26, 2023

(Click on the comic if you can't see the full image.)
(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

They go in any which direction they wish.

Except when it's raining.

This is one of those jokes that when it occurs to me, I find myself wondering how I didn't think of it 15 years ago. Better late than never. And the later you get there, the more the meter has run up, which is only fare.

### I also write Fiction!

You can now order Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
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. ## Friday, July 21, 2023

### Tall-People Problems

(Click on the comic if you can't see the full image.)
(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

My pants get wet, and my shoes get squishy.

When your umbrella is higher than everyone else's, your runoff falls onto their umbrellas. And their runoff falls onto you.

### I also write Fiction!

You can now order Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
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. ## Thursday, July 20, 2023

### June 2023 Geometry Regents Part IV

This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

### June 2023 Geometry, Part IV

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

35. Given: Quadrilateral ABCD, AB ≅ CD, AB || CD, diagonal AC intersects EF at G, and DE ≅ BF

Prove: G is the midpoint of EF

I basically called this question "evil". Why? Because of several things that were put in just to trip you up. Okay, so I was being a bit overdramatic, but there are quite a few things that need to be stated. You should have at least 10 steps in your proof. (I could have combined two of my statements and used the reason "Opposite sides in a parallelogram are both parallel and congruent.")

For starters, you are given that ABCD is a "quadrilateral". You are not given that it is a parallelogram. So before you do anything else, you're going to need to show that it is, because you will need those properties later.

To prove that G is the midpoint of EF, you need to show that EG ≅ GF. This means that you need to show that two triangles are congruent so that their corresponding parts will be (CPCTC).

The two traingles that must be congruent are AGE and CGF. If we first prove that ABCD is a parallelogram, we can show that AE ≅ FC, and use alternate interior angles and vertical angles to set up AAS.

Your proof should look like this:

 Statement Reasons AB ≅ CD, AB || CD, diagonal AC intersects EF at G, and DE ≅ BF Given ABCD is a parallelogram A quadrilateral with one pair of sides that are parallel and congruent is a parallelogram AD ≅ BC The opposite sides of a parallelogram are congruent. AE ≅ CF Subtraction Postulate Angle AGE ≅ angle CGF Vertical Angles AD || CB Opposite sides of a parallelogram are parallel Angle EAG ≅ angle FCG Alternate interior angles Triangle AGE ≅ triangle CGF AAS EG ≅ FG CPCTC G is the midpoint of EF If a point divides a segment into two congruent parts, it must be the midpoint.

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.

End of Exam

How did you do?

### I also write Fiction!

You can now order Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder 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 1 Regents Part IV

This exam was adminstered in June 2023. These answers were not posted until after the Independence Holiday weekend, well after the school year ednded.

More Regents problems.

### June 2023

Part IV: Each correct answer will receive 6 credits. Partial credit can be earned.

37. Dana went shopping for plants to put in her garden. She bought three roses and two daisies for \$31.88. Later that day, she went back and bought two roses and one daisy for \$18.92.

If r represents the cost of one rose and d represents the cost of one daisy, write a system of equations that models this situation. Use your system of equations to algebraically determine both the cost of one rose and the cost of one daisy.

If Dana had waited until the plants were on sale, she would have paid \$4.50 for each rose and \$6.50 for each daisy. Determine the total amount of money she would have saved by buying all of her flowers during the sale.

Write a system of equations. Remember to use r and d, not x and y. It will be solved the same way.

The second sentence tells us 3r + 2d cost 31.88. The third sentence says that 2r + 1d cost 18.92. So:

3r + 2d = 31.88
2r + d = 18.92

You can use either elimination or substition to solve this. To use substitution, rewrite the second equation as

d = -2r + 18.92
and then substitute it into the first equation.
3r + 2(-2r + 18.92) = 31.88
and solve for r.

Or use elimination, by multiplying the second equation by 2.

3r + 2d = 31.88
2r + d = 18.92

3r + 2d = 31.88
4r + 2d = 37.84

-r= -5.96
r = 5.96

1 rose costs \$5.96.

2(5.96) + d = 18.92
11.92 + d = 18.92
d = 7

1 daisy costs \$7.00.

The final portion of the question could be answered even if you didn't solve the system of equations. It could be solved even if you didn't WRITE the system of equations.

You already know how much money Dana spent. It's in the problem. Find how much she would have spent with the flowers on sale, and then subtract the two numbers.

(31.88 + 18.92) - (5*(4.50) + 3(6.50)) = \$8.80

She would've saved \$8.80.

As far as Part IV problems go, this one wasn't particularly difficult. It was basically a Part III question with an additional problem tacked onto the end of it.

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 Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder 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.  ## Tuesday, July 18, 2023

### June 2023 Algebra 1 Regents Part III

This exam was adminstered in June 2023. These answers were not posted until after the Independence Holiday weekend, well after the school year ednded.

More Regents problems.

### June 2023

Part III: Each correct answer will receive 4 credits. Partial credit can be earned.

33. Graph f(x) = |x| + 1 and g(x) = -x2 + 6x + 1 on the set of axes below.
Based on your graph, determine all values of x for which f(x) = g(x).

Put both of these equations into your graphing calculator and check the table of values. Make sure you either draw off the edge of the chart, or include arrows at your endpoints. And draw enough to show all the points of intersection. And remember to label at least one of the lines.

Your graph should look like this:

The values of x where f(x) = g(x) are 0 and 5.

Do NOT write coordinates. They asked for x values.

34. Jean recorded temperatures over a 24-hour period one day in August in Syracuse, NY. Her results are shown in the table below.

Her data are modeled on the graph below.
State the entire interval over which the temperature is increasing.
State the three-hour interval that has the greatest rate of change in temperature.
State the average rate of change from hour 12 to hour 24. Explain what this means in the context of the problem.

Increasing means where there's a positive slope on the graph. The graph is increasing between hour 6 and hour 12. You could write it as [6,12] or 6 < x < 12, or just write it in words from 6 to 12. Whether it is an open or closed interval is debatable at this level, so either answer would've been accepted. College professors are free to argue otherwise in the comments, or to write to NY State and tell them that they are big doody heads.

The interval with the greatest rate of change would be the same as the one with the steepest slope. It is obvious from the graph, but you could measure the rise over run, that the hours from 9 to 12 have the greatest change. Note that it doesn't say positive change. However 9 to 12 is steeper in a positive direction than any of the negative slopes in the chart.

The average rate of change from hours 12 to 24 is the slope between (12,92) and (24,74).
(74 - 92) / (24 - 12) = -18/12 = -3/2.
In the context of this problem, the temperature was dropping 3 degrees every 2 hours.

35.Solve the following system of inequalities graphically on the set of axes below.
2x + 3y > -6
x < 3y + 6

Label the solution set S.

Is the point (4,-2) in the solution set?

If you can graph the first inequality, which is in Standard Form, great! If not, rewrite it along with the second one in slope-intercept form so you can graph it.

2x + 3y > -6
3y > -2x - 6
y > -2/3 x - 2

x < 3y + 6
-3y < -x + 6
y > 1/3 x - 2

Notes: Both lines will have y-intercept at (0,-2). The first inequality has a solid line with a negative slope. The second inequality will have a broken (or dashed) line and a positive slope. Both inequalities will be shaded above their lines.

Your graph will look like this. The S will go in the top section, which is shaded twice with the crisscross pattern. Remember to label at least one of the lines, and use the original inequalities, not the ones you put in your calculator.

No, (4,-2) in not in the solution set. It is not in the section label S. It is in a section that is only a solution to one of the inequalities. (Phrase it however you like.)

NOTE: If you make a graphing error, such as shading below the dashed line, then (4,-2) may be in the solution set of your graph. If so, then your answer will be Yes. You will not be penalized twice. (You will lose a point for the graphing error.)

36. Suzanna collected information about a group of ponies and horses. She made a table showing the height, measured in hands (hh), and the weight, measured in pounds (lbs), of each pony and horse.
Write the linear regression equation for this set of data. Round all values to the nearest hundredth.
State the correlation coefficient for the linear regression. Round your answer to the nearest hundredth.
Explain what the correlation coefficient indicates about the linear fit of the data in the context of the problem.

Enter the data into Lists L1 and L2 on your calculator. Run the linear regression. If you didn't get values for r and r2, then you need to run the command DiagonsticsOn from the Catalog. You will need the r value.

When you run the linear regression, you will get a = 184.89 and b = 1706.07, to the nearest hundredth.
So the equation will be y = 184.89x - 1706.07.

The correlation coefficient is 0.99, which is the value of r to two decimal places.

The context is this: As the height of the horse increases, the weight of the horse increases.

If you said that there's a strong positive correlation, that would be true, but it wouldn't state the context of the problem.

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 Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder 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 Geometry Regents Part III

This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

### June 2023 Geometry, Part III

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

31. Cape Canaveral, Florida is where NASA launches rockets into space. As modeled in the diagram below, a person views the launch of a rocket from observation area A, 3280 feet away from launch pad B. After launch, the rocket was sighted at C with an angle of elevation of 15o. The rocket was later sighted at D with an angle of elevation of 31o.

Determine and state, to the nearest foot, the distance the rocket traveled between the two sightings, C and D.

To find the length of CD, you need to find the length of BD and BC and then subtract BD - BC. In both cases, you have the angle and the adjacent side and you are looking for the opposite side, so you want to use tangent ratios.

tan 15 = x/3280, so x = 3280 * tan 15 = 878.873..

tan 31 = y/3280, so y = 3280 * tan 31 = 1970.822..

CD = 1970.822 - 878.873 = 1091.949 = 1092 feet.

33. TA small can of soup is a right circular cylinder with a base diameter of 7 cm and a height of 9 cm. A large container is also a right circular cylinder with a base diameter of 9 cm and a height of 13 cm. Determine and state the volume of the small can and the volume of the large container to the nearest cubic centimeter.

What is the minimum number of small cans that must be opened to fill the large container? Justify your answer.

I can't tell you (because even if I remembered, I couldn't disclose it) the number of students who missed up the second part of the question.

The question gives diameters, but the formula uses radius. Halve each number.

V(sm) = π (3.5)2(9) = 346.36... = 346 cm3

V(lg) = π (4.5)2(13) = 827.024... = 827 cm3

For the second part, divide the volume of the larger can by the volume of the smaller can. And then round UP. If you don't round up, you won't fill the larger can.

827/346 = 2.39..., so you would need to open 3 cans.

34. Parallelogram MATH has vertices M(-7,-2), A(0,4), T(9,2), and H(2,-4).
Prove that parallelogram MATH is a rhombus.
[The use of the set of axes below is optional.]

Determine and state the area of MATH.

You probably want to plot the points just to make it easier to visualize, but it isn't necessary.

You must either show that the four sides have the same length, or you must show that the diagonals are perpendicular and bisect each other. It is not enough to show that the diagonals are perpendicular because the diagonals of a kite are perpendicular also.

MA: √( (-7 - 0)2 + (-2 - 4)2 ) = √ (85)

AT: √( (0 - 9)2 + (4 - 2)2 ) = √ (85)

TH: √( (9 - 2)2 + (2 - -4)2 ) = √ (85)

MA: √( (2 - -7)2 + (-4 - -2)2 ) = √ (85)

All four sides are congruent, so MATH is a rhombus. This concluding statment is required.

The easiest way to find the area of the rhombus is to draw a rectangle around it that has points M, A, T, and H on each of its sides. Four triangles will be formed. Find the area of the rectangle and subtract the areas of the four triangles.

You have the numbers that you need -- you found the rise and run when you did the distance formula!

The rectangle has an area of 16 * 8 = 128. The four triangles have areas of 1/2(6)(7) = 21, 1/2(9)(2) = 9, 1/2(6)(7) = 21, and 1/2(9)(2) = 9.

So 128 - (21 + 9 + 21 + 9) = 68

You could also have found the lengths of the two diagonals and multiplied 1/2 d1d2.

End of Part III

How did you do?

### I also write Fiction!

You can now order Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder 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.  ## Wednesday, July 05, 2023

### June 2023 Geometry Regents Part II

This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

### June 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. In △ABC below, use a compass and straightedge to construct the altitude from C to AB.
[Leave all construction marks.]

An altitude is a perpendicular line, and the way to create one is to make a perpendicular bisector. But first we need a segment with two endpoints to bisect. Do NOT use A and B.

The first step is to put the compass on point C. Open it wide enough to swing past line AB in two places. These two points will be used for your perpendicular bisector.

Since we already have point C, you only have to make arcs below the line to find another point. If you make them above the line, that is okay, because they will line up with point C.

Draw a line from point C to the point below the line you just found. That line will contain the altitude of the triangle.

Look at this image. I didn't use my own because "faking" construction in MS Paint is not an easy task, and it sometimes doesn't come out just right.

26. Triangles ABC and DEF are graphed on the set of axes below.

Describe a sequence of transformations that maps △ABC onto △DEF.

This was an odd question because a "sequence" is not needed. It is one simple rotation. No translations, reflections or dilations needed afterward.

A rotation of 90 degrees clockwise about the origin maps △ABC onto △DEF.

You need to say rotation, how much, and about which center to get both credits.

You also could've said, for instance, a rotation of 90 degrees clockwise (or 270 CCW) around point A or point C, and then listed the required translation to move onto DEF.

27. Line segment PQ has endpoints P(–5,1) and Q(5,6), and point R is on PQ. Determine and state the coordinates of R, such that PR:RQ = 2:3.
[The use of the set of axes below is optional.]

If the ratio is 2:3 that PR is 2/5 of the length of line segment and RQ is 3/5.

Find the difference between the two x-coordinates. Multiply that number by 2/5. Then add the result to -5. Do the same for the two y-coordinates. Find the differnce. Multiply by 2/5, but add the result to 1, because P is at (-5,1).

5 - -5 = 10; 2/5(10) = 4. -5 + 4 = -1. The x-coordinate of R is -1. (If you plot the line, you can see what the y value will be.)

6 - 1 = 5. 2/5(5) = 2. 1 + 2 = 3. The y-coordinate of R is 3. The location of R is (-1.3)

If you plot P and Q and draw the line with a straightedge, you will see that the line goes through the point (-1,3).

28. A circle has a radius of 6.4 inches. Determine and state, to the nearest square inch, the area of a sector whose arc measures 80°.

The area of a circle is πr2. The area of a sector is the area of the full circle times the measure of the central angle divided by 360 degrees.

A = (80/360)π(6.4)2 = 28.5955...

The area is approximately 29 square inches.

29. A large snowman is made of three spherical snowballs with radii of 1 foot, 2 feet, and 3 feet, respectively. Determine and state the amount of snow, in cubic feet, that is used to make the snowman.

The formula for the Volume of a sphere is 4/3 π r3. You will need to use the formula 3 times and then add the results together.

Do NOT use the value of pi or press the pi key on your calculator when calculating.

V = 4/3 π (1)3 + 4/3 π (1)3 + 4/3 π (1)3

V = 4/3 π ( (1)3 + (2)3 + (3)3)

V = 4/3 π (1 + 8 + 27)

V = 4/3 π (36)

V = 48 π

Interesting note: 1 + 2 + 3 = 6, and (1)3 + (2)3 + (3)3 = 62

30. In the diagram below of right triangle ACB, altitude CD is drawn to hypotenuse AB, AD = 2 and AC = 6.

Determine and state the length of AB.

Some students look at this and see the Right Triangle Altitude Theorem, except we don't immediately know the altitude. You can find it with Pythagorean Theorem, but remember to leave the answer as a radical. Do NOT round it, because an error will creep into your final answer.

The large right triangle is divided into two smaller right triangles. All three triangles are similar, and their corresponding angles are congruent. This means that their corresponding sides are proportional.

In triangle ACD, we know the hypotenuse and the short leg.

In triangle ABC, we know the short leg and we are looking for the hypotenuse.

Set up a proportion:

Short/hyp = short/hyp>
2 / 6 = 6 / AB
2 AB = 36
AB = 18

AB has a length of 18.

If you used the Pythagorean Theorem, you would've found that CD has a length of √(32). Then DB would equal 16. Finally, AB = AD + DB = 2 + 16 = 18. Same result.

31. Triangle RST has vertices with coordinates R(-3,-2), S(3,2) and T(4,-4). Determine and state an equation of the line parallel to RT that passes through point S. [The use of the set of axes below is optional.]

You can graph it but it isn't likely to help, except to tell you that between what two numbers the y-intercept is, but you can't read it precisely because it's a fraction.

First, find the slope of RT. Then use that slope and point S to come up with the equation of the line. If you remember Point-Slope Form, it's not to difficult. If you use Slope-Intercept Form, it's a little trickier because of the fractions. Use fractions. Don't round decimals!

The slope of RT = (-4 - -2) / (4 - -3) = -2/7, which isn't a very nice fraction.

Point-Slope Form, using S: y - 2 = -2/7(x - 3)

Slope-Intercept Form:

y = mx + b
2 = (-2/7)(3) + b
2 = -6/7 + b
20/7 = b
y = -2/7 x + 20/7

20/7 could be written as a mixed number, 2 6/7, but not as a decimal because it's infinite.

End of Part II

How did you do?

### I also write Fiction!

You can now order Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder 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 1 Regents Part II

This exam was adminstered in June 2023. These answers were not posted until after the Independence Holiday weekend, well after the school year ednded.

More Regents problems.

### 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. Solve the equation algebraically for x:

-2.4(x + 1.4) = 6.8x - 22.68

You have the choice of using the Distributive Property first, or dividing both sides of the equation by -2.4. In this case, dividing by -2.4 will lead to infinite decimals, and you don't want to round in the middle of the problem, so let's use the Distributive Property.

-2.4(x + 1.4) = 6.8x - 22.68
-2.4x - 3.36 = 6.8x - 22.68
19.32 = 9.2x
x = 2.1

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

State the zeroes of f(x).

The zeroes of the function are the values of x that make f(x) = 0. In other words, it's the values on the x-axis where the graph crossing the x-axis (which is the line y = 0 or f(x) = 0).

The zeroes are -2, 2, and 3.

I gave a reason above, but you could write something like those are the values of the x-intercepts.

Do NOT write points/coordinates, such as (-2,0). That isn't what they asked for.

The number 12 is the y-intercept. It is not a correct answer.

A more interesting question -- and the one that I expected -- was to give a possible equation for this function.

The definition of the function is f(x) = a(x + 2)(x - 2)(x - 3), but what is the value of a?

When x = 0, according to the graph f(0) = 12. And (0 + 2)(0 - 2)(0 - 3) = 12. So a = 1.

So the function is f(x) = (x + 2)(x - 2)(x - 3).

27. Breanna creates the pattern of blocks below in her art class.

A friend tells her that the number of blocks in the pattern is increasing exponentially.
Is her friend correct?

The friend is not correct. It is increasing by 2, a constant rate of change, or a common difference, which means it's a linear pattern not and exponential one.

28. The data set 20, 36, 52, 56, 24, 16, 40, 4, 28 represents the number of books purchased by nine book club members in a year.
Construct a box plot for these data on the number line below.

You can use the calculator to sort the data, and even have it give you the Five Number Summary that you need to construct the box-and-whisker plot.

If you don't know how to do that, you can do it by hand since there are only 9 data items.

The numbers is order are:

4, 16, 20, 24, 28, 36, 40, 52, 56

The mminimum number is 4 and the maximum number is 56. There are nine numbers, so the fifth number is the one in the center, the median, which is 28.

There are four numbers less than the median. Q1 is halfway between the 2nd and 3rd numbers. Halfway between 16 and 20 is 18. Likewise, Q3 is halfway between the 7th and 8th numbers. Halfway between 40 and 52 is 46.

So the Five-Number Summary, which you should write down, is: Min = 4, Q1 = 18, Med = 28, Q3 = 46 and Max = 56.

Plot points above these positions on the number line. Note that the scale has already been set. Then draw a box that goes through Q1 and Q3 and add a vertical line through the median. Finally, draw the two whiskers from Q1 to the Min and Q3 to the Max.

29. Given: A = x + 5 and B = x2 - 18

Express A2 + B in standard form.

Substitute each expression. Square the binomial (x + 5). Combine like terms. Rewrite in Standard form with the term with the highest exponent going first.

(x + 5)2 + x2 - 18
x2 + 10x + 25 + x2 - 18
2x2 + 10x + 7

That's it.

30. The two relations shown below are not functions.

Explain how you could change each relation so that they each become a function.

Important: they don't want you to just tell them why it is not a function. You MUST state something that would fix the relation, making it a function.

In Relation 1: you could make (4,20) an open circle, or you could make (4,30) an open circle, or you could even remove the entire middle segment from 2 < x < 4.

In Relation 2: you CANNOT say "change one of the -4's to another number". That is not specific enough. Plus, it isn't entirely true. There are other numbers that you are not allowed to change it to if you want the result to be a function.

In Relation 2, you could say, remove (-4, 0), or change (-4,4) to (-3,4), or something specific like that. If you change the x value for x = -4, do NOT change it to a number that is already there.

31. Factor completely: 2x2 + 16x - 18

"Factor completely" almost always means multiple steps. Remove the Common Factor (2) from each term and then factor the trinomial into two binomials.

2x2 + 16x - 18 = 2(x2 + 8x - 9) = 2(x + 9)(x - 1)

32. Solve 3d2 - 8d + 3 = 0 algebraically for all values of d, rounding to the nearest tenth.

The fact that it has to be rounded to the nearest tenth usually means that you want to use the Quadratic Formula because you might not be able to factor it otherwise.

These numbers look like they would work out with the "borrow and payback" method, or the reverse box method, or whatever you may have heard it called. However, they will not. You need to factors of +9 with a sum of -8. You can't do that with integers. So let's use the Quadratic Formula.

Follow along on the image below:

Rounded, the solutions are d = 2.2 and d = 0.5.

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 Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder 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.  ## Tuesday, July 04, 2023

### Happy Independence Day 2023

(Click on the comic if you can't see the full image.)
(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

You have the freedom to solve this or not to.

The answer is left as an excerise for whatever app you math students use to do your homework for you.

### I also write Fiction!

You can now order Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
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. 