2.5: Practice Problems
- Page ID
- 139267
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1. A function 𝑃(𝑛) gives the wholesale price in dollars of a package of cookies after n packages have been sold. Interpret the meaning of 𝑃(124)=0.8.
2. The function 𝐶(𝑔) represents the cost to produce g gallons of chocolate chunk ice cream. Interpret the meaning of 𝐶(751)=160.
3. The function 𝐷(𝑡) can be used to approximate the total average credit card debt in a U.S. household (in thousands of dollars) t years after 1995. Interpret the meaning of 𝐷(29)=21.5.
4. A function 𝑑(ℎ) gives the number of miles Billy has driven in his car from his house h hours after leaving his house.
a. Interpret 𝑑(4)=160
b. Interpret 𝑑(0.1)=2
5. The function 𝑔(𝑡) is given by the table below.
|
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
|
g(t) |
12 |
24 |
3 |
24 |
27 |
0 |
3 |
ta. Evaluate 𝑔(1)
b. Evaluate 𝑔(6)
c. Determine t when 𝑔(𝑡)=24
6. The function 𝑓(𝑥) is represented below as a graph. Use 𝑓(𝑥) to answer the following questions:
a. Evaluate 𝑓(−1)
b. Evaluate 𝑓(3)
c. Determine x when 𝑓(𝑥)=0
d. Determine x when 𝑓(𝑥)=6
7. The function 𝑓(𝑥) is represented below as a graph. Use 𝑓(𝑥) to answer the following questions:
a. Evaluate 𝑓(0)
b. Evaluate 𝑓(4)
c. Determine x when 𝑓(𝑥)=−3
d. Determine x when 𝑓(𝑥)=1
8. Given the function 𝑔(𝑥)=𝑥+2
a. Evaluate 𝑔(−5)
b. Evaluate 𝑔(0)
c. Determine x when 𝑔(𝑥)=−2
9. Given the function 𝑘(𝑡)=−8𝑡+9
a. Evaluate 𝑘(3)
b. Evaluate 𝑘(0)
c. Determine t when 𝑘(𝑡)=−39
10. Identify the rate of change and initial value for the function 𝑓(𝑥)=3𝑥+7
11. Identify the rate of change and initial value for the function 𝑓(𝑥)=−9−10𝑥
12. Identify the rate of change and initial value for the function 𝑓(𝑥)=−5𝑥
13. Identify the rate of change and initial value for the function 𝑓(𝑥)=1
14. Calculate the rate of change of the linear function that contains the points (1,8) and (4,17).
15. Calculate the rate of change of the linear function that contains the points (−15,30) and (7,−14).
16. Calculate the rate of change of the linear function that contains the points (3,2) and (10,2).
17. Determine the rate of change and initial value of the linear function that generates the following table of values.
|
x |
-4 |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
|
f(x) |
-70 |
-49 |
-28 |
-7 |
14 |
35 |
56 |
77 |
98 |
18. Determine the rate of change and initial value of the linear function that generates the following table of values.
|
|
-4 |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
|
f(x) |
102 |
75 |
48 |
21 |
-6 |
-33 |
-60 |
-87 |
-114 |
19. Determine the rate of change and initial value of the linear function that generates the following table of values.
|
|
-4 |
-2 |
0 |
2 |
4 |
|
f(x) |
-90 |
-14 |
62 |
138 |
214 |
20. The function 𝑉(𝑡)=−5.4𝑡+34.2 gives the value (in thousands of dollars) of an investment after t years. Interpret the rate of change in this situation.
21. The function 𝑉(𝑥)=2.5𝑥+25.4 gives the value (in thousands of dollars) of an investment after x months. Interpret the rate of change in this situation.
22. Paul is planning to sell bottled water at the local carnival. Paul's profit (in dollars) from selling b bottles of water is given by the formula 𝑃(𝑏)=1.4𝑏−338. Interpret the rate of change in this situation.
23. When a new charter school opened in 1996, there were 440 students enrolled. Write a formula for the function 𝑁(𝑡), representing the number of students attending this charter school t years after 1996, assuming that the student population:
a. Increased by 44 students per year
b. Decreased by 32 students per year
c. Remained constant (did not change)
24. A town's population has been growing linearly. In 2003, the population was 59,000 and the population has been growing by 1,700 people each year.
a. Write a formula for the population x years after 2003.
b. What will the town’s population be in 2007?
c. In what year will the population be 77,700 people?
25. Last year, Pinwheel Industries introduced a new toy. It cost $1,300 to develop the toy and $30 to manufacture each toy.
a. Write a formula for the total cost to produce n of these toys.
b. What is the total cost to produce 4,400 toys?
c. How many total toys can be produced with $124,300?
26. In the year 1999, the surface elevation of Lake Powell was 3848 feet above sea level. In the year 2007, the surface elevation of Lake Powell was 3,227.2 feet above sea level. Find the rate of change of the surface elevation during this time span.
27. In the year 1987, an investment was worth $29,800. In the year 1997, this investment was worth $43,800.
a. Find the rate of change of the investment during this time span.
b. Is the investment increasing or decreasing?
28. In the year 1982, an investment was worth $26,200. In the year 1990, this investment was worth $39,000.
a. Find the rate of change of the investment during this time span.
b. Is the investment increasing or decreasing?
29. In 1995, the cost of tuition at a large Midwestern university was $170 per credit hour. In 2000, tuition had risen to $235 per credit hour.
a. Find the rate of change of the cost of tuition during this time span.
b. Write a formula for the cost of tuition as a function of the number of years since 1990.
c. What will the tuition be in 2007?
d. What year will the tuition be $417 per credit hour?
30. Robert has $100 in his school lunch account. If the school charges $3.50 per lunch, how many lunches can he purchase? How much money will remain in his account?
31. You are interested in starting a gym membership and have narrowed your selection to two gyms. The first, Phoenix Fitness, charges a one-time membership fee of $99 and charges $35 per month. The second, Slim Gym, charges a one-time membership fee of $10 and charges $40 per month.
a. Which gym is the more affordable choice if you are going to commit to a 1-year membership?
b. Which gym is the more affordable choice if you are going to commit to a 2-year membership?
32 a. Use your graphing calculator to create of scatterplot of the data set shown below.
|
|
1 |
3 |
4 |
6 |
7 |
9 |
10 |
|
|
437 |
901 |
1155 |
1358 |
1768 |
2103 |
2437 |
b. Does the data set appear to be increasing or decreasing?
c. Does the data set appear to be roughly linear?
33. a. Use your graphing calculator to create of scatterplot of the data set shown below.
|
|
2 |
9 |
14 |
23 |
33 |
42 |
|
|
-60.2 |
-130.1 |
-243.7 |
-328.9 |
-580.5 |
-643.8 |
b. Does the data set appear to be increasing or decreasing?
c. Does the data set appear to be roughly linear?
34. a. Use your graphing calculator to create of scatterplot of the data set shown below.
|
|
8 |
12 |
14 |
23 |
24 |
29 |
33 |
|
|
10 |
20 |
15 |
24 |
35 |
30 |
28 |
b. Does the data set appear to be increasing or decreasing?
c. Does the data set appear to be roughly linear?
35. a. Use your graphing calculator to create of scatterplot of the data set shown below.
|
|
20 |
24 |
27 |
32 |
48 |
60 |
68 |
|
|
12 |
20 |
28 |
41 |
24 |
15 |
11 |
b. Does the data set appear to be increasing or decreasing?
c. Does the data set appear to be roughly linear?
36. Calculate the linear regression equation for the data from problem 32 :
|
|
1 |
3 |
4 |
6 |
7 |
9 |
10 |
|
|
437 |
901 |
1155 |
1358 |
1768 |
2103 |
2437 |
37 Calculate the linear regression equation for the data from problem 33 :
|
|
2 |
9 |
14 |
23 |
33 |
42 |
|
|
-60.2 |
-130.1 |
-243.7 |
-328.9 |
-580.5 |
-643.8 |
38 Calculate the linear regression equation for the data from problem 34 :
|
|
8 |
12 |
14 |
23 |
24 |
29 |
33 |
|
|
10 |
20 |
15 |
24 |
35 |
30 |
28 |
39. Consider the following 4 scatterplots. Match each scatterplot with the most appropriate value from the 4 correlations listed below.
r = 0.38 r = 0.94 r = -0.15 r = -0.85
40. Find and interpret the linear correlation for the data from problem 32:
|
|
1 |
3 |
4 |
6 |
7 |
9 |
10 |
|
|
437 |
901 |
1155 |
1358 |
1768 |
2103 |
2437 |
41. Find and interpret the linear correlation for the data from problem 33:
|
x |
2 |
9 |
14 |
23 |
33 |
42 |
|
y |
-60.2 |
-130.1 |
-243.7 |
-328.9 |
-580.5 |
-643.8 |
42. Find and interpret the linear correlation for the data from problem 34:
|
x |
8 |
12 |
14 |
23 |
24 |
29 |
33 |
|
|
10 |
20 |
15 |
24 |
35 |
30 |
28 |
43. Find and interpret the linear correlation for the data from problem 35:
|
|
20 |
24 |
27 |
32 |
48 |
60 |
68 |
|
|
12 |
20 |
28 |
41 |
24 |
15 |
11 |
44. The following table shows the number of newspaper subscriptions in Middletown, USA where t represents the number of years since 2002 (t = 0 in 2002) and S(t) represents the total subscriptions each year measured in thousands.
|
|
0 |
2 |
4 |
6 |
8 |
|
S(t) (total subscriptions in 1000s) |
448 |
372 |
198 |
145 |
45 |
Find and interpret the linear correlation.
45. Scott is hiking the Appalachian Trail from Georgia to Maine. The distance of his hike is 2200 miles. It took Scott 123 days to complete the hike. The data below represent the distance, D, he had hiked t days after the start of his trip.
|
t (days hiking) |
0 |
32 |
47 |
73 |
99 |
123 |
|
D(t) distance in miles |
0 |
590 |
912 |
1212 |
1876 |
2200 |
Find and interpret the linear correlation.
46. Which implies a stronger linear relationship, a correlation of -0.84 or -0.47? Explain.
47. Which implies a stronger linear relationship, a correlation of -0.73 or 0.51? Explain.
48. The following table shows the number of newspaper subscriptions in Middletown, USA where t represents the number of years since 2002 (t = 0 in 2002) and S(t) represents the total subscriptions each year measured in thousands.
|
|
0 |
2 |
4 |
6 |
8 |
|
S(t) (total subscriptions in 1000s) |
448 |
372 |
198 |
145 |
45 |
a) Use your graphing calculator to create of scatterplot of the data set.
b) Based on your graph above, do the data appear to be approximately linear?
c) Use your calculator to determine the regression equation in S(t) = at + b form. (Round to 2 decimal places)
d) What is the slope of your regression model for S(t) and what is its meaning in the context of this problem?
e) What is the initial value of your linear regression model for S(t) and what is its meaning in the context of the problem.
f) Use your linear regression equation to estimate the total number of subscriptions in 2007 (i.e. when t =5). Show your computations here and your final result.
g) Use your linear regression equation to estimate the total number of subscriptions in 2004. How does this value compare to the data value in the table?
h) Should you use your linear regression equation to estimate the circulation in the year 2030? Why or why not?
49. Scott is hiking the Appalachian Trail from Georgia to Maine. The distance of his hike is 2200 miles. It took Scott 123 days to complete the hike. The data below represent the distance, D, he had hiked t days after the start of his trip.
|
t (days hiking) |
0 |
32 |
47 |
73 |
99 |
123 |
|
D(t) distance in miles |
0 |
590 |
912 |
1212 |
1876 |
2200 |
a) Use your graphing calculator to create of scatterplot of the data set.
b) Based on your graph above, do the data appear to be approximately linear?
c) Use your calculator to determine the regression equation in D(t) = at + b form. (Round to 2 decimal places)
d) What is the slope of your regression model for D(t) and what is its meaning in the context of this problem?
e) Use your linear regression equation to estimate the total number of miles Scott has hiked in 50 days. Show your computations here and your final result.
f) Should you use your linear regression equation to estimate when Scott has hiked 2500 miles? Why or why not?