June 2018 Common Core Algebra I Regents, Parts III and IV

The following are some of the multiple questions from the recent June 2018 New York State Common Core Algebra I Regents exam.
The answers to Part I can be found here
The answers to Part II can be found here

June 2018 Algebra I, Part III

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

33. A population of rabbits in a lab, p(x), can be modeled by the function p(x) = 20(1.014)^x, where x represents the number of days since the population was first counted.
Explain what 20 and 1.014 represent in the context of the problem.
Determine, to the nearest tenth, the average rate of change from day 50 to day 100.

Answer:
In the context of the problem, 20 is the initial population of the rabbits and 1.014 is the rate of growth of the rabbit population of 1.4% per day.
1.014 = 101.4%, which is 1.4% increase.

To find the average rate of change from day 50 to day 100, find p(50) and p(100). Subtract them and divide by 100 - 50 = 50.
p(50) = 20(1.014)⁵⁰ = 40.08...
p(100) = 20(1.014)¹⁰⁰ = 80.32...
(80.32 - 40.08) / (100 - 50) = 40.24/50 = 0.8048. which is approximately 0.8 to the nearest tenth.
The rate of change is 0.8.

34. There are two parking garages in Beacon Falls. Garage A charges $7.00 to park for the first 2 hours, and each additional hour costs $3.00. Garage B charges $3.25 per hour to park.
When a person parks for at least 2 hours, write equations to model the cost of parking for a total of x hours in Garage A and Garage B.
Determine algebraically the number of hours when the cost of parking at both garages will be the same.

Answer:
A: y = 3.00(x - 2) + 7.00 ($7 for the first 2 hours. Those 2 hours are subtracted from x)
B: y = 3.25x (All hours are the same.)

3.25x = 3.00(x - 2) + 7.00
3.25x = 3.00x - 6.00 + 7.00
.25x = 1.00
x = 4
They will cost the same after 4 hours.

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

2y + 3x < 14
4x - y < 2
Determine if the point (1,2) is in the solution set. Explain your answer.

Answer:
Note that the y term is first in the first inequality.
If you are rewriting these into slope-intercept form to put them into your calculator, be mindful of the inequality symbol.

2y + 3x < 14 2y < -3x + 14 y < (-3/2)x + 7 negative slope, solid line, shaded below

4x - y < 2 - y < -4x + 2 y > 4x - 2 positive slope, broken line, shaded above (left)

See image below.
The point (1, 2) is NOT in the solution set because it is on a broken line, which is not part of the solution.

36. The percentage of students scoring 85 or better on a mathematics final exam and an English final exam during a recent school year for seven schools is shown in the table below.

Percentage of Students Scoring 85 or Better
Mathematics, x	English, y
27	46
12	28
13	45
10	34
30	56
45	67
20	42

Write the linear regression equation for these data, rounding all values to the nearest hundredth.
State the correlation coefficient of the linear regression equation, to the nearest hundredth. Explain the meaning of this value in the context of these data.

Answer:
Put all the data into the first two lists on your graphing calculator. Make sure you have DiagnosticsOn (so you'll get the correlation coefficient), and run a linear regression. Round all values to TWO decimals (nearest hundredth).
You will get the equation y = .96x + 23.95
The correlation coefficient is r = .92.
In the context of these data, there is a strong positive correlation between the percentage of students scoring over 85 in Mathematics and English. As one increases, the other increases.

End of Part III

June 2018 Algebra I, Part IV

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

37. Dylan has a bank that sorts coins as they are dropped into it. A panel on the front displays the total number of coins inside as well as the total value of these coins. The panel shows 90 coins with a value of $17.55 inside of the bank.
If Dylan only collects dimes and quarters, write a system of equations in two variables or an equation in one variable that could be used to model this situation.
Using your equation or system of equations, algebraically determine the number of quarters Dylan has in his bank.
Dylan’s mom told him that she would replace each one of his dimes with a quarter. If he uses all of his coins, determine if Dylan would then have enough money to buy a game priced at $20.98 if he must also pay an 8% sales tax. Justify your answer.

Answer:
Let D be the number of dimes and Q be the number of quarters.
The value of the dimes will be .10D and the value of the quarters will be .25Q.
The equations you need to write to model this situation are:

D + Q = 90
.10D + .25Q = 17.55

To find the number of quarters, solve the system of equations.

D + Q = 90
.10D + .25Q = 17.55
-------------------
-.10D - .10Q = -9.0
.10D + .25Q = 17.55
-------------------
.15Q = 8.55
Q = 57
Dylan has 57 quarters in his bank.
Check:
D + 57 = 90, D = 90 - 57 = 33
.10(33) + .25(57) = 3.30 + 14.25 = 17.55 (check)

If his mom replaces all the dimes with quarters, then Dylan has 90 quarters, which are worth 90*.25= $22.50
The game costs 20.98 + 20.98(0.08) = 22.6584 or $22.66.
Dylan doesn't have enough money. He will be 16 cents short. (or $.16, but NOT ".16 cents")
You could also multiply 20.98 times 1.08, which is the price plus 8%.

End of Part IV

How did you do?

Questions, comments and corrections welcome.