August 2017 Common Core Algebra 1 Regents, Part 2

I recently realized that students are reviewing the August 2017 Algebra 1 Regents, and I never wrote up the open-ended problems.
Here are the Part I questions, answers, and explanations.

August 2017, Algebra 1 (Common Core), Part II

25. A teacher wrote the following set of numbers on the board:

Explain why a + b is irrational, but b + c is rational.

Answer: The sum a + b is irrational because the sum of an irrational number and a rational number is always irrational.
The sum of b + c is rational because the square root of 225 is 15, which is a rational number. The sum of two rational numbers is always rational.

26. Determine and state whether the sequence 1, 3, 9, 27,… displays exponential behavior. Explain. how you arrived at your decision.

Answer: The sequence displays exponential behavior because it has a common ratio.
3 / 1 = 3
9 / 3 = 3
27 / 9 = 3

27. Using the formula for the volume of a cone, express r in terms of V, h, and π. how you arrived at your decision.

Answer:

V = 1/3 π r ² h
3 V = π r ² h
3 V / (πh) = r ²
square root of (3 V / (πh)) = r

28. The graph below models the cost of renting video games with a membership in Plan A and Plan B.

Explain why Plan B is the better choice for Dylan if he only has $50 to spend on video games, including a membership fee.

Bobby wants to spend $65 on video games, including a membership fee. Which plan should he choose? Explain your answer.

Answer: According to the graph, Bobby can get 14 games under Plan B but he can only get 12 games under Plan A.
According to the graph, if Bobby plans to spend $65, then both plans will give him the same number of games because that is the point where the two plans are the same.

29. Samantha purchases a package of sugar cookies. The nutrition label states that each serving size of 3 cookies contains 160 Calories. Samantha creates the graph below showing the number of cookies eaten and the number of Calories consumed.

Explain why it is appropriate for Samantha to draw a line through the points on the graph.

Answer: Samantha should draw a line through the points on the graph because she could eat only 1 or 2 cookies, or even a part of a cookie.

30. A two-inch-long grasshopper can jump a horizontal distance of 40 inches. An athlete, who is five feet nine, wants to cover a distance of one mile by jumping. If this person could jump at the same ratio of body-length to jump-length as the grasshopper, determine, to the nearest jump, how many jumps it would take this athlete to jump one mile.

Commentary: I'm surprised that they didn't include the unit conversions necessary to complete this problem. However, they are in the back of the test book.

Answer: The grasshopper is 2 inches and can jump 40 inches, which is a ratio of 40:2, or 20 times his body length.
The person is 5'9", which converts to 5.75 feet (because 9/12 of a foot is .75 feet). Multiply this by 20: 5.75 * 20 = 115 feet per jump.
One mile is 5280 feet, so 5280 / 115 = 45.9, which rounds to 46 jumps would be needed.

If you got as far as finding 115 feet (or 1380 inches), you would have gotten one point. You need to get to 46 jumps, rounded correctly, to get the other.

31. Write the expression 5x + 4x²(2x + 7) - 6x² - 9x as a polynomial in standard form.

Answer: Use the Distributive Property, and then Combine Like Terms
Standard form means the term with the highest exponent goes first and then in decreasing order.
Remember that the sign goes with the term after it, and the first term here is positive (+).

5x + 4x²(2x + 7) - 6x² - 9x
5x + 8x³ + 28x² - 6x² - 9x
8x³ + 28x² - 6x² + 5x - 9x
8x³ + 22x² - 4x

32. Solve the equation x² - 6x = 15 by completing the square

Answer: b = -6, so b / 2 = -3, which means that (x - 3) will be in the answer.
Also, (b / 2)² = 9. Add 9 to both sides to begin.

x² - 6x + 9 = 15 + 9
(x - 3)² = 24
x - 3 = +(24)^(1/2)
x = 3 +(24)^(1/2)

It isn't necessary to reduce (radical (24)) into (2 X radical (6)).

End of Part II.
How did you do?
Corrections, comments, questions are welcome.