January 2016 New York Algebra I (Common Core) Parts 3 and 4

Below are the questions with answers and explanations for Part 3 and 4 of the Algebra I (Common Core) Regents exam for January 2016. The multiple-choice questions appeared in a previous post. Part II questions appeared in a another post.

As always, in order to get this thread up quickly, the images have been omitted. They will be added at a later date.

Each question in Part 3 is worth 4 credits, for a total of 16 credits. Partial credit will be given. The one question in Part 4 is worth 6 credits.

January 2016 Algebra 1 (Common Core) Regents, Part 3

33. Let h(t) = -16t² + 64t + 80 represent the height of an object above the ground after t seconds. Determine the number of seconds it takes to achieve its maximum height. Justify your answer.

State the time interval, in seconds, during which the height of the object decreases. Explain your reasoning.

The maximum height is at the vertex, and the vertex is on the axis of symmetry.
The axis of symmetry is x = -b/(2a) = -64/(2(-16)) = -64/-32 = 2 seconds.
h(2) = -16(2)² + 64(2) + 80 = -16(4) + 128 + 80 = -64 + 128 + 80 = 144. It will be 144 feet off the ground.

It will decrease in height from 2 seconds until it hits the ground, which is when h(t) = 0.
Solve -16t² + 64t + 80 = 0
Divide by -16: t² - 4t - 5 = 0
Factor: (t - 5)(t + 1) = 0
Solve t = 5 or t = -1.
Discard the negative time value. The object hits the ground at 5 seconds.

The object is descending from 2 < t < 5.
At 5 seconds, it is no longer descending. Before 2, it was still rising.

34. Fred's teacher gave the class the quadratic function f(x) = 4x² + 16x + 9.
a) State two different methods Fred could use to solve the equation f(x) = 0.
b) Using one of the methods stated in part a, solve f(x) = 0 for x, to the nearest tenth.

a) Pick any two methods for solving are factoring, completing the square and the quadratic formula. There is also graphing, but that won't be helpful if the answer is not an integer.

Note that factoring may not be possible. If it turns out that it isn't, cross that one out and use the other two.
4x² + 16x + 9 = 0 -- 4 and 9 are the squares of 2 and 3 and (2)(2)(3) = 12, not 16, so it is (2x + 3)². (One could hope it would be easy.)

Personally, I prefer the Quadratic formula is Completing the Square is going to involve fractions anyway, so look at the illustration:

So the solutions are {-3.3, -.7}

35. Erica, the manager at Stellarbeans, collected data on the daily high temperature and revenue from coffee sales. Data from nine days this past fall are shown in the table below. (image omitted)

State the linear regression function, f(t) that estimates the day's coffee sales with a high temperature of t. Round all values to the nearest integer.

State the correlation coefficient, r, of the data to the nearest hundredth. Does r indicate a strong linear relationship between the variables? Explain your reasoning.

You need to put the data into LISTs in the graphing calculator and then do a Linear Regression. Before the exam, your calculator should have had its memory reset. After that, the two things that should have happened were that it was put back into Degree Mode, and DiagnosticsON should have been executed. It's a function in the CATALOG. With that on, the correlation coefficient will appear on the screen when you run a Linear Regression.

Put the temperatures in L1, and the sales in L2. Double check your work. (I had one typo, and that would have skewed my answer!)
Hit Stat, go to the Calc menu, and press 4: Linear Regression. Press ENTER.

f(t) = -58t + 6182 -- use f(t) and t. Don't use y and x.

The correlation coefficient is -.94 (to the nearest hundredth). This indicates a strong negative relationship because the number is close to -1, meaning that it is almost a straight line.

36. A contractor has 48 meters of fencing that he is going to use as the perimeter of a rectangular garden. The length of one side of the garden is represented by x, and the area of the garden is 108 square meters.

Determine, algebraically, the dimensions of the garden in meters.

P = 2x + 2w = 48; 2w = 48 - 2x; w = 24 - x
A = L * W = x (24 - x) = 108
24x - x² = 108
0 = x² - 24x + 108
0 = (x - 6)(x - 18)
x = 6 or x = 18

The dimensions of the garden are 6 m X 18 m.
Check: 6 * 18 = 108. 2(6) + 2(18) = 12 + 36 = 48. Check.

January 2016 Algebra 1 (Common Core) Regents, Part 4

37. The Reel Good Cinema is conducting a mathematical study. In its theater, there are 200 seats. Adult tickets cost $12.50 and child tickets cost $6.25. The cinema's goal is to sell at least $1500 worth of tickets for the theater.

Write a system of linear inequalities that can be used to find the possible combinations of adult tickets, x, and child tickets, y, that would satisfy the cinema's goals.

Graph the solution to this system of inequalities on the set of axes on the next page. Label the solution S.

Marta claims that selling 30 adult tickets and 80 child tickets will result in meeting the cinema's goal. Explain whether she is correct or incorrect, based on the graph drawn.

The graph will be coming shortly. Please be patient.

The system of inequalities is:

x + y < 200
12.50x + 6.25y >1500

When you graph them, both will have solid lines. The first inequality (# of tickets) will be shaded below. The second inequality (money from the sales) will be shaded above the line. The area shaded twice will be get the S.

Marta is incorrect. (If you graphed correctly) If you look at the graph, (30, 80) is not in the section with the S, so it is not a solution to the system of inequalities. (You needed to refer to the graph, so just plugging the numbers into both inequalities is not sufficient.)

EDIT: Graph added

End of Test

How'd you do?