(x, why?) Mini: A Root and a Fifth

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

I hope the joke didn't fall flat.

I'm not very musically inclined, but I was recently informed that a root and a fifth make a power chord. How could I note, er, not?

I also write Fiction! You can now preorder 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.

Come back often for more funny math and geeky comics.

Algebra Problems of the Day (Integrated Algebra Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part II: Each correct answer will receive 2 credits. Partial credit is possible.


31. Solve the following system of equations algebraically for y:

2x + 2y = 9
2x - y = 3

Answer:

You only need to find y, not x. Both equations have 2x in them so they can be subtracted to eliminate the variable. Remember that you have to subtract a negative y.

2x + 2y = 9
2x - y = 3
3y = 6
y = 2

You can check your work by substituting y = 2 into each equation and seeing if you get the same value of x for both. If you don't, you made a mistake.

2x + 2(2) = 9
2x + 4 = 9
2x = 5
x = 2.5

2x - 2 = 3
2x = 5
x = 2.5

The answer checks, so y = 2.

32. Three storage bins contain colored blocks. Bin 1 contains 15 red and 14 blue blocks. Bin 2 contains 16 white and 15 blue blocks. Bin 3 contains 15 red and 15 white blocks. All of the blocks from the three bins are placed into one box.

If one block is randomly selected from the box, which color block would most likely be picked? Justify your answer.

Answer:

I remember when this question first appeared. I thought it was dumb then, and I still do. It has an unnecessary complication. My only guess is that it was supposed to be a more difficult problem, but they simplified it to use for Part II instead. That much is only a guess.

If all the contents are poured into one bin, you have to total the number of red, white and blue blocks:

Red: 15 + 0 + 15 = 30

White: 0 + 16 + 15 = 31

Blue: 14 + 15 + 0 = 29

Total: 30 + 31 + 29 = 90 (You actually don't need this because you aren't asked for an exact probability.)

You need to write something like: "A white block is most likely to be picked because there are more white blocks in the bin than any other color."

33. Students calculated the area of a playing field to be 8,100 square feet. The actual area of the field is 7,678.5 square feet. Find the relative error in the area, to the nearest thousandth.

Answer:

Reminder: relative error is NOT percent of error, even though the two are absolutely related. (One is a decimal and one is a percentage.) If you put a percentage, you will lose half credit because they didn't ask for a percentage.

Also, "to the nearest thousandth" means three decimal places, properly rounded.

Relative error is | actual - estimated | / actual. Find the difference between the two amounts and divide it by the actual amount.

| 8,100 - 7,678.5| / 7,678.5 = 0.05489..., which is 0.055 to the nearest thousandth.

End of Part II.

More to come. Comments and questions welcome.

More Regents problems.

Algebra Problems of the Day (Integrated Algebra Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part I: Each correct answer will receive 2 credits.


26. When 8x² + 3x + 2 is subtracted from 9x² - 3x - 4, the result is

1) x² - 2
2) 17x² - 2
3) -x² + 6x + 6
4) x² - 6x - 6

Answer: 4) x² - 6x - 6

You take the first expression away from the second expresion. in other words, it's the second expression minus the first one. You may want to write it vertically.

(9x² - 3x - 4) - (8x² + 3x + 2)
9x² - 3x - 4 - 8x² - 3x - 2
9x² - 8x² - 3x - 3x - 2 - 4
x² - 6x - 6

27. Factored completely, the expression 3x³ - 33x² + 90x is equivalent to

1) 3x(x² - 33x + 90)
2) 3x(x² - 11x + 30)
3) 3x(x + 5)(x + 6)
4) 3x(x - 5)(x - 6)

Answer: 4) 3x(x - 5)(x - 6)

It should be obvious that each term is a multiple of 3 and each term has a factor of x. Of course, this is a "no-brainer" since each of the four choice includes 3x. That said, divide each term by 3x and then see if you can factor the rest.

3x³ - 33x² + 90x = 3x(x² - 11x + 30x) Eliminate Choice (1).

However, x² - 11x + 30x can be factored into (x - 5)(x - 6). Which gives you Choice (4).

28. Elizabeth is baking chocolate chip cookies. A single batch uses 3/4 teaspoon of vanilla. If Elizabeth is mixing the ingredients for five batches at the same time, how many tablespoons of vanilla will she use?

3 teaspoons = 1 tablespon

1) 1 1/4
2) 1 3/4
3) 3 3/4
4) 5 3/4

Answer: 1) 1 1/4

Find the total number of teaspoons needed, then convert that to tablespoons.

3/4 * 5 = 15/4 teaspoons * 1 tablespoon / 3 teaspoons = 15/12 tablespoons.

15/12 = 1 3/12 = 1 1/4

29. A car depreciates (loses value) at a rate of 4.5% annually. Greg purchased a car for $12,500. Which equation can be used to determine the value of the car, V, after 5 years?

1) V = 12,500(0.55)⁵
2) V = 12,500(0.955)⁵
3) V = 12,500(1.045)⁵
4) V = 12,500(1.45)⁵

Answer: 2) V = 12,500(0.955)⁵

If it is LOSING value than the rate must be less that 1.00 or 100%. Eliminate Choices (3) and (4).

You subtract the amount, 4.5% from 100%, which is 95.5% or .955. That is Choice (2).

Choice (1) would be if the appreciation was 45% instead of 4.5%.

30. The cumulative frequency table below shows the length of time that 30 students spent text messaging on a weekend.
Which 10-minute interval contains the first quartile?

1) 31 - 40
2) 41 - 50
3) 51 - 60
4) 61 - 70

Answer: 3) 51 - 60

The median would be 15.5, and the lower half of the data would be the bottom 15 numbers. The median on the lower half (which would be the lower quartile) would be the 8th student. The 8th student would be in the 31-60 interval on the cumulative frequency table, meaning that the first quartile is contained in the interval of 51-60.

More to come. Comments and questions welcome.

More Regents problems.

Algebra Problems of the Day (Integrated Algebra Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part I: Each correct answer will receive 2 credits.


21. Which equation is represented by the graph below?

1) 2y + x = 10
2) y - 2x = -5
3) -2y = 10x - 4
4) 2y = -4x - 10

Answer: 4) 2y = -4x - 10

You can pick a couple of points (especially the y-intercept) from the graph and try them in the equations. Only one equation can have two points as solutions because two points define a line.

2y + x = 10, 2(-5) + 0 =/= 10. Eliminate Choice (1).

y - 2x = -5, -5 - 2(0) = -5. Try a second point.
(-4, 3): 3 - 2(-4) =/= -5. Eliminate Choice (2).

-2y = 10x - 4, -2(-5) =/= 10(0) - 4. Eliminate Choice (3).

2y = -4x - 10, 2(-5) = -4(0) - 10.
(-4, 3): 2(3) ?= -4(-4) - 10
6 = 16 - 10 = 6. Choice (4) is the answer.

The alternative is to find the equation of the line and see which of the choices it resembles.

The slope is -2 rise and 1 run, or m = -2. The y-intercept is -5, so the equation for this line is y = -2x - 5, which is 2x + y = -5 in Standard form, which is NOT the same as (2).

If you multiply the equation by 2, you get 4x + 2y = -10, which then becomes 2y = -4x - 10.

22. Which coordinates represent a point in the solution set of the system of inequalities shown below?

y < 1/2x + 13
4x + 2y > 3

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

Answer: 4) (2, -2)

Pick either of the two inequalities and try the points out.

4(-4) + 2(1) = -13, which is not greater than 3. Eliminate Choice (1).

4(-2) + 2(2) = -4, which is not greater than 3. Eliminate Choice (2).

4(1) + 2(-4) = -4, which is not greater than 3. Eliminate Choice (3).

4(2) + 2(-2) = 4 > 3.
3 ?< 1/2 (2) + 13
3 < 14. Choice (4) is the answer.

23.The length of one side of a square is 13 feet. What is the length, to the nearest foot, of a diagonal of the square?

1) 13
2) 18
3) 19
4) 26

Answer: 2) 18

If you don't know the shortcurt, use the Pythagorean Theorem. However, at the very least, you know that you can eliminate 13, because it has to be longer, and 26, because 13-13-26 does NOT make a triangle.

a² + b² = c²
13² + 13² = c²
169 + 169 = c²
338 = c²
c = 18.38...

The diagonal, to the nearest foot, is 18.

The shortcut is realizing that two sides of a square and a diagonal make an isosceles right triangle where the hypotenuse is the length of one leg times the square root of two.

13 * sqrt(2) = 18.38...

24. In triangle ABC, m∠C = 90. If AB = 5 and AC = 4, which statement is not true?

1) cos A = 4/5
2) tan A = 3/4
3) sin B = 4/5
4) tan B = 5/3

Answer: 4) tan B = 5/3

If the right triangle has a leg of 4 and hypotenuse of 5, then the other leg is 3. We should show this with Pythagorean Theorem, but really, by this point, you should just KNOW that 3-4-5 is the simplest, smallest Pythagorean Triple.

The rules for sin, cos and tan are: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.

Cos A = adj / hyp = 4/5

Tan A = opp / adj = 3/4

Sin B = opp B / hyp = 4/5, which is the same as Cos A.

Tan B = opp B / adj B = 4/3, not 5/3. The hypotenuse is 5, but it is NOT used in tangents.

25. If n is an odd integer, which equation can be used to find three consecutive odd integers whose sum is -3?

1) n + (n + 1) + (n + 3) = -3
2) n + (n + 1) + (n + 2) = -3
3) n + (n + 2) + (n + 4) = -3
4) n + (n + 2) + (n + 4) = -3

Answer: 3) n + (n + 2) + (n + 4) = -3

If n is an odd number, then the next odd number is n + 2 and the next one after that is n + 4. The sum of them equals -3.

More to come. Comments and questions welcome.

More Regents problems.

(x, why?) Mini: What's Your Function?

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

I almost wrote How do you even fnction? But let's not get into Even functions right now.

When I see pictures of polar graphs, I think of maps of Antartica where everything on the perimeter is North.

I also write Fiction! You can now preorder 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.

Come back often for more funny math and geeky comics.

Algebra Problems of the Day (Integrated Algebra Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part I: Each correct answer will receive 2 credits.


16. Which expression represents (x² - 3x - 10) / (x² - 25) in simplest form?

1) 2 / 5
2) (x + 2) / (x + 5)
3) (x - 2) / (x - 5)
4) (-3x - 10) / -25

Answer: 2) (x + 2) / (x + 5)

Factor both the numerator and the denominator into two binomials. Then cancel out the common factors that form a multiplicative identity. (That is N / N = 1)

(x² - 3x - 10) / (x² - 25)
( (x - 5) (x + 2) ) / ( (x - 5) (x + 5) )
(x + 2) / (x + 5)

You cannot cancel the x² terms in the original expression because they are not FACTORS. There are multiple terms being added and subtracted. Likewise, you cannot cancel the x terms in the final answer.

17. Which interval notation describes the set S = {x |1 < x < 10}?

1) [1,10]
2) (1,10]
3) [1,10)
4) (1,10)

Answer: 3) [1,10)

Interval notation is shorthand for a compound inequality that uses less than and less than or equal to. A rounded parenthesis means "less than" but not equal to, and a square bracket means use "less than or equal to". Note that Choice (4) is NOT a point on the coordinate plane, even though it looks like one. This is because mathematicians used to do everything by hand, so this notation saved a lot of space and ink.

The lower boundary has < so the left symbol will be a bracket. The upper boundary is <, so it gets a parenthesis.

18.The bull’s-eye of a dartboard has a radius of 2 inches and the entire board has a radius of 9 inches, as shown in the diagram below

If a dart is thrown and hits the board, what is the probability that the dart will land in the bull’s-eye?

1) 2/9
2) 7/9
3) 4/81
4) 49/81

Answer: 3) 4/81

Find probability by dividing the amount of positive results by the ammount all possible results. Wit a darboard, we need to compare the areas of the bull's-eye with the area of the entire board. That means that we have to find the Area of two circles and divide them.

Note that because we will divide them, we don't need to multiply by pi, because the two factors of pi will cancel.

Area of bull's-eye = (2)² * pi = 4 * pi

Area of dartboard = (9)² * pi = 81 * pi

(4 pi) / (81 pi) = 4 / 81

19. What is one-third of 3⁶?

1) 1²
2) 3²
3) 3⁵
4) 9⁶

Answer: 3) 3⁵

One-third of 3⁶ is 1/3 * 3⁶ = 3⁶ / 3 = 3⁶ / 3¹.

The rules of exponents say that you subtract the exponents: 6 - 1 = 5

The Axis of Symmetry is x = -b / (2a) = -24 / (2(-2)) = 6

20. The expression (2x + 13) / (2x + 6) - (3x - 6) / (2x + 6) is equivalent to

1) (-x + 19) / (2(x + 3))
2) (-x + 7) / (2(x + 3))
3) (5x + 19) / (2(x + 3))
4) (7x + 7) / (4x + 12)

Answer: 1) (-x + 19) / (2(x + 3))

You can only add fractions when the denominators are the same, and then the denominator stays the same. You do NOT add the denominators, so Choice (4) is eliminated.

(2x + 13) / (2x + 6) - (3x - 6) / (2x + 6)
( (2x + 13) - (3x - 6) ) / (2x + 6)
( 2x + 13 - 3x + 6) / (2x + 6)
( -x + 19 ) / (2x + 6)

More to come. Comments and questions welcome.

More Regents problems.

Algebra Problems of the Day (Integrated Algebra Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part I: Each correct answer will receive 2 credits.


11. If five times a number is less than 55, what is the greatest possible integer value of the number?

1) 12
2) 11
3) 10
4) 9

Answer: 3) 10

If five times a number is less than 55 than one times the number is less than 11. But it is NOT 11.

12. The line represented by the equation 2y - 3x = 4 has a slope of

1) -3/2
2) 2
3) 3
4) 3/2

Answer: 4) 3/2

Don't get fooled by the y term being written first. It isn't written in Standard Form, in case you want to use a formula that required standard form.

You can rewrite the equation, solving for y:

2y - 3x = 4
2y = 3x + 4
y = 3/2 x + 2

Or you can switch around the terms and use the formula:

2y - 3x = 4
-3x + 2y = 4
Ax + By = C
Slope equals -A / B = -(-3) / 2 = 3/2

13. What is the solution set of the system of equations x + y = 5 and y = x² - 25?

1) {(0,5), (11, -6)}
2) {(5,0), (-6,11)}
3) {(-5,0), (6,11)}
4) {(-5,10), (6,-1)}

Answer: 2) {(5,0), (-6,11)}

You can solve the systems of equations, or you can check the solutions to see which sets have two points that satisfy both equations.

If you checking the solutions, start with x + y = 5 to see if anything can be eliminated, because it's the easiest to deal with

0 + 5 = 5, 11 + -6 = 5

5 + 0 = 5, -6 + 11 = 5

-5 + 0 =/= 5, Eliminate Choice (3)

-5 + 10 = 5, 6 + -1 = 5

Next try the other equation. Remember, you can skip Choice (3):

0² - 25 =/= 5, Eliminate Choice (1)

5² - 25 = 0, (-6)² - 25 = 36 - 25 = 11. Check. Choice (2) is the solution.

(-5)² - 25 =/= 10. Eliminate Choice (4).

14. What is the vertex of the parabola represented by the equation y = -2x² + 24x - 100?

1) x = -6
2) x = 6
3) (6, -28)
4) (-6, -316)

Answer: 3) (6, -28)

The vertex of a parabola is a point, not a line. So eliminate Choices (1) and (2). One of those might be the Axis of Symmetry which will go through the vertex, but neither is the vertex.

The Axis of Symmetry is x = -b / (2a) = -24 / (2(-2)) = 6

The y-coordinate of the vertex is y = -2(6)² + 24(6) - 100 = -28. So (6, -28)

You also could have found this by putting the equation in your graphing calcuator and looking at the table of values to find the turning point. Note that since (6, -28) is a MAXIMUM, you wouldn't see anything on your screen unless you changed the scale shown in the window.

15. If k = am + 3mx, the value of m in terms of a, k, and x can be expressed as

1) k / (a + 3x)
2) (k - 3mx) / a
3) (k - am) / 3x
4) (k - a) / 3x

Answer: 1) k / (a + 3x)

If you are solving for m in terms of a, k and x, then you will NOT have m in the solution. You can eliminate (2) and (3) immediately.

Solve for m by using Inverse Operations to isolate the variable:

k = am + 3mx
k = m(a + 3x)
k / (a + 3x) = m

More to come. Comments and questions welcome.

More Regents problems.

Young Man

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Based on a true story.

If you're younger, you're younger. Who's to say what's "young" any way?

You can set up a compound inequality if you like.

I also write Fiction! 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. Thank you.

Come back often for more funny math and geeky comics.

Algebra Problems of the Day (Integrated Algebra Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part I: Each correct answer will receive 2 credits.


6. Which situation is an example of bivariate data?

1) the number of pizzas Tanya eats during her years in high school
2) the number of times Ezra puts air in his bicycle tires during the summer
3) the number of home runs Elias hits per game and the number of hours he practices baseball
4) the number of hours Nellie studies for her mathematics tests during the first half of the school year

Answer: 3) the number of home runs Elias hits per game and the number of hours he practices baseball

Bivariate means that there are two pieces of data being tracked and compared. In Choice (3), there is data for Elias scoring hits and for the hours he practices. These can be compared.

In Choice (1), only the number of pizzas is being counted.

In Choice (2), just the number of times air in put into tires is tracked.

In Choice (4), the hours studying for tests is measured, but not, say, the scores on those tests.

7. Brianna’s score on a national math assessment exceeded the scores of 95,000 of the 125,000 students who took the assessment. What was her percentile rank?

1) 6
2) 24
3) 31
4) 76

Answer: 4) 76

Given the choices provided, you don't have to do much math at all. If you beat 95,000 of 125,000, you are well above the halfway point. I could do the math to see if that would be 76 percent, but the other three numbers are so low, why bother. Choices (1), (2) and (3) are just silly.

Anyway, 95000 / 125000 = 0.76, which is the 76th percentile.

8. 8 If A = {0, 1, 3, 4, 6, 7}, B = {0, 2, 3, 5, 6}, and C = {0, 1, 4, 6, 7}, then A ∩ B ∩ C is

1) {0, 1, 2, 3, 4, 5, 6, 7}
2) {0, 3, 6}
3) {0, 6}
4) {0}

Answer: 3) {0, 6}

The intersection of two sets in the set of all elements that are present in both the first and the second set. The intersection of three sets is the set of all elements that are present in every one of the three individual sets. You can do one intersection at a time to make it less confusing, if you want to.

The intersection A ∩ B = {0, 1, 3, 4, 6, 7} ∩ {0, 2, 3, 5, 6} = {0, 3, 6}

The intersection A ∩ B ∩ C = {0, 3, 6} ∩ {0, 1, 4, 6, 7} = {0, 6}

9.Which graph represents a function?

Answer: 1)

For a graph to be a function, it must pass the vertical-line test, which states that no vertical line can be drawn through any part of the graph so that it touches more than one point on the graph.

In Choice (2), a vertical line can be drawn between, for example, (2, 2) and (2, -2). The same x-value (the input) has two different y-values (output). This is not allowed in a function.

Similarly, Choices (3) and (4) fail the test.

10. What is the product of (3x + 2) and (x - 7)?

1) 3x² - 14
2) 3x² - 5x - 14
3) 3x² - 19x - 14
4) 3x² - 23x - 14

Answer: 3) 3x² - 19x - 14

If the four choices, the first and last terms are the same. You only need to concern yourself with the middle term.

If you use the Area Model, you can see that the four products that you need to find are:

(3x)(x) = 3x²

(3x)(-7) = -21x

(2)(x) = 2x

(2)(-7) = -14

When you combine the like terms, you see that -21x + 2x = -19x, which is Choice (3).

More to come. Comments and questions welcome.

More Regents problems.

Algebra Problems of the Day (Integrated Algebra Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part I: Each correct answer will receive 2 credits.


1. In a baseball game, the ball traveled 350.7 feet in 4.2 seconds. What was the average speed of the ball, in feet per second?
1) 83.5
2) 177.5
3) 354.9
4) 1,472.9

Answer: 1) 83.5

To find the unit rate of speed in feet per second divide the number of feet by the amount of seconds:

350.7 / 4.2 = 83.5

Common sense should tell you that if something travels a little over 350 feet in a little over 4 seconds, then in just ONE second, it should go about 1/4 of the distance. Choices (3) and (4) are just ridiculous. Choice (2) isn't much better, but at least it's smaller.

2. A survey is being conducted to determine if a cable company should add another sports channel to their schedule. Which random survey would be the least biased?
1) surveying 30 men at a gym
2) surveying 45 people at a mall
3) surveying 50 fans at a football game
4) surveying 20 members of a high school soccer team

Answer: 2) surveying 45 people at a mall

Bias means that you are more likely to find people who are predeisposed to give one particular answer over another. It also comes from focusing too much on one particular subgroup of people.

In Choice (1), the survey would only be asking men of their opinion, and in particular, men who are most likely athletic.

In Choice (3), the survey is specifically asking fans if they want more sports while they are attending a sporting event.

In Choice (4), the survey is again asking people involved with sports, but this time they are all of high school age, and there aren't even that many of them.

The only problem with Choice (2) is that 45 people may not seem like very many, but you will find a more random collection of people to ask.

3. The quotient of (8x⁵ - 2x⁴ + 4x³ - 6x²) / (2x²) is
1) 16x⁷ - 4x⁴ + 8x⁵ - 12x⁴
2) 4x⁷ - x⁶ + 2x⁵ - 3x⁵
3) 4x³ - x² + 2x - 3x
4) 4x³ - x² + 2x - 3

Answer: 4) 4x³ - x² + 2x - 3

Divide each coefficient by 2 and reduce each exponent by 2. An exponent of 1 can be left out because it means one factor of x, which is just x. An exponent of 0 means that there are NO factors of the variable, so leave the variable off.

If should be obvious that Choice (1) is silly. It's the only one that even looks like that. Despite the word "quotient" and the fraction in the problem, Choice (1) shows the product of the two expressions.

Choice (2) is similarly wrong because it divided the coefficients but it added the expoents, instead of reducing them.

Choice (3) is incorrect because it ends with "2x - 3x", which you should wonder, why weren't those like terms combined? Because they weren't supposed to be like terms. If the original exponents were 5, 4, 3, 2, and each is reduced by 2, then you should have 3, 2, 1, 0 in your answer. But -3x has an exponent of 1.

Choice (4) is correct because -3x⁰ is -3.

4. Marcy determined that her father’s age is four less than three times her age. If x represents Marcy’s age, which expression represents her father’s age?
1) 3x - 4
2) 3(x - 4)
3) 4x - 3
4) 4 - 3x

Answer: 1) 3x - 4

Three times her age, x, is 3x. Four less than that means take four away from that or subtract 4 from that amount. That is 3x - 4.

4 - 3x is "obviously" wrong because her father would be less than 4 years old. In fact, if Marcy was over the age of 1, her father's age would be a negative number.

5. A set of data is graphed on the scatter plot below.
This scatter plots shows

1) no correlation
2) positive correlation
3) negative correlation
4) undefined correlation

Answer: 2) positive correlation

As the x values get bigger, the y values generally get bigger. If you were to draw a line of best fit through the data, it would have a positive slope.

More to come. Comments and questions welcome.

More Regents problems.

Algebra Problems of the Day (Integrated Algebra Regents, August 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part IV: Each correct answer will receive 4 credits. Partial credit is possible.


37. A company is running a contest and offering a first, second, and third prize. First prize is a choice of a car or $15,000 cash. Second prize is a choice of a motorbike, a trip to New York City, or $2,000 cash. Third prize is a choice of a television or $500 cash.

If each prize is equally likely to be selected, list the sample space or draw a tree diagram of all possible different outcomes of first, second, and third prizes.

Determine the number of ways that all three prizes selected could be cash.

Determine the number of ways that none of the three prizes selected could be cash.

Answer:

The sample space would be (you can abbreviate, but be clear about it):
Car, motorbike, TV
Car, motorbike, $500
Car, NYC, TV
Car, NYC, $500
Car, $2000, TV
Car, $2000, $500
$15K, motorbike, TV
$15K, motorbike, $500
$15K, NYC, TV
$15K, NYC, $500
$15K, $2000, TV
$15K, $2000, $500

The tree diagram would look like this:

The tree diagram or sample space is considered the WORK for the next two questions. If you answer the questions based on what you drew, you can get credit even if you made a mistake in your diagram, as long as it is consistent.

If you did NOT create a tree diagram or sample space, then work is expected to be shown on the bottom two questions, such as using the Counting Principle.

There is only ONE way that all three prizes will be cash. ($15,000, $2,000, $500)

There are TWO ways that none of the three prizes will be cash. (Car, bike, TV, and Car, NYC, TV)

38. In right triangle ABC shown below, AC = 29 inches, AB = 17 inches, and m∠ABC = 90. Find the number of degrees in the measure of angle BAC, to the nearest degree.

Find the length of BC to the nearest inch.

Answer:

You are given that it is a right triangle, along with the length of the hypotenuse and one of the sides. You want to find the size of the angle ADJACENT to the side you know. That means using Cos.

Cos x = 17/29
x = cos^-1(17/29) = 54.11...

The measure of ∠BAC is 54 degrees.

To find the length of BC, you have a choice of using the Sine ratio, or Pythagorean Theorem.

17² + BC² = 29²
289 + BC² = 841
BC² = 552
BC = 23.49...
BC = 23 inches

Or...

sin 54.11 = BC/29
BC = 29 * sin 54.11 = 23.49...
BC = 23 inches

39.On the set of axes below, graph the following system of inequalities.

y + x ≥ 3
5x - 2y > 10

State the coordinates of one point that satisfies y + x ≥ 3, but does not satisfy 5x - 2y > 10.

Answer:

Rewrite the inequalities in slope-intercept form, so you can use your calculator, or just plot from the y-intercept and the slope.

y + x ≥ 3
y ≥ -x + 3

5x - 2y > 10
- 2y > -5x + 10
y < 5/2 x - 5

Remember to shade on the correct sides. Remember that the second inequality is a broken or dashed line.

You may choose any point that is in the top quarter of the graph, the top of the "X", for the final answer. You may choose a point on the top part of the red line, the top part of the blue line, or any point in between them. For example, (2, 2).

End of Exam.

More to come. Comments and questions welcome.

More Regents problems.

Algebra Problems of the Day (Integrated Algebra Regents, August 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part III: Each correct answer will receive 3 credits. Partial credit is possible.


34. The following cumulative frequency histogram shows the distances swimmers completed in a recent swim test.

Based on the cumulative frequency histogram, determine the number of swimmers who swam between 200 and 249 yards.

Determine the number of swimmers who swam between 150 and 199 yards.

Determine the number of swimmers who took the swim test.

Answer:

There are three questions and each is worth a credit.

For the first question, to find the number of swimmers who swam between 200 and 249 yards, look at the columns labeled "50–249" and "50–299". The difference between those two amounts is what you are looking for. Subtract 20 - 15 = 5. There were 5 swimmers in that range.

Similarly, to find the number between 150 and 199, look at " 50–149" and "50–199". They are both the same, so 12 - 12 = 0. There were 0 swimmers in that range.

The number of swimmers is 20 because the cumulative frequency chart counts everyone, and everyone will be included in the final column.

35. Ashley measured the dimensions of a rectangular prism to be 6 cm by 10 cm by 1.5 cm. The actual dimensions are 5.9 cm by 10.3 cm by 1.7 cm. Determine the relative error, to the nearest thousandth, in calculating the volume of the prism.

Answer:

Reminder: Relative error is a decimal. Percent of error is a percentage. Should that make a difference? Honestly, I don't think so, but the people who write the rubrics would disagree with me. And their voice is the one that matters, unfortuately.

What this means is, if you do all the work correctly and then express the answer as a correct percentage, you will lose a point, which amounts to 1/3 of the credit for the problem. Or about 33.3%.

To find the relative error, calculate the volume using Ashley's measurements. Then calculate the actual volume with the actual measurements. Find the difference between those two amounts and divide it by the actual volume. Round it to three decimal places.

Ashley's volume: V = L * W * H = 6 * 10 * 1.5 = 90

Actual volume: V = L * W * H = 5.9 * 10.3 * 1.7 = 103.309

Error: (103.309 - 90) / (103.309) = 0.1288...

To the nearest thousandth, the relative error is 0.129.

36. Solve the following system of equations algebraically for all values of x and y.

y = x² + 2x - 8
y = 2x + 1

Answer:

Note that you could graph it or use a table of values, but in doing so, you would only get one point for a correct answer for not solving it algebraically. You can also do this first if you want to know what the answer is before you start solving it algebraically. (There's nothing wrong with doing this.)

To solve the system of equations, set the two equations equal to each other. Since each expression is equation to y, then they are equal to each other. Once you do that, you can solve the quadratic equation that you get.

y = x² + 2x - 8
y = 2x + 1
x² + 2x - 8 = 2x + 1
x² - 9 = 0
(x + 3)(x - 3) = 0
x + 3 = 0 or x - 3 = 0
x = -3 or x = 3

Now that you know the values of x, go back and find the matching values of y.

y = 2(-3) + 1 = -6 + 1 = -5. (-3, -5)

y = 2(3) + 1 = 6 + 1 = 7. (3, 7)

End of Part III.

More to come. Comments and questions welcome.

More Regents problems.