Download Quadratic Applications

MA 097 (Scott)
Quadratic Models (Chp.7)
Show work for 1 - 4. Use the appropriate features of your graphing calculator to answer 5 - 8.
SHOW WORK neatly and legibly on YOUR PAPER.
1) A manufacturer determines that the profit in dollars for manufacturing n units is P = 2n2 - 60 n - 400 .
(Assume that n is a positive integer) How many units are produced when the profit is $400?
2) April shoots an arrow upward into the air at a speed of 32 feet per second from a platform that is 28 feet
high. The height of the arrow is given by the function h(t) = -16t2 + 32t + 28, where t is the time is seconds.
What is the maximum height of the arrow? Use the vertex formula.
3) A ball is thrown upward with an initial velocity of 14 meters per second from a cliff that is 90 meters high.
The height of the ball is given by the quadratic equation h = -4.9t2 + 14t + 110 where h is in meters and t is
the time in seconds since the ball was thrown. After how many seconds will the height of the ball be 20
meters? Round your answer to the nearest tenth of a second. Use the Quadratic Formula.
Find the vertex, the y-intercept, and the x-intercepts (if any exist), and graph the function.
4) y = -x2 - 2x + 8
Use a graphing calculator to plot the data and find the quadratic function of best fit.
5) A small manufacturing firm collected the following data on advertising expenditures (in thousands of
dollars) and total revenue (in thousands of dollars).
Advertising, x Total Revenue, R
6430
25
6432
28
6434
31
6434
32
6434
34
6431
39
6432
40
6420
45
Find the quadratic function of best fit.
6) The following table shows the median number of hours of leisure time per week for Americans in various
years.
Year
1973 1980 1987 1993 1997
Median Number of Leisure Hours per Week 26.2
19.2
16.6
18.8
19.5
Use x = 0 to represent the year 1973. Using a graphing utility, determine the quadratic regression equation
for the data given. In which year was the median amount of leisure time the lowest?
1
Solve the problem.
7) The median weekly incomes of full-time male and female workers for various age groups are listed in the
table below.
Age
Income (in dollars)
Group Women Men
16-24
331
377
25-34
478
593
35-44
514
713
45-54
558
780
55-65
490
728
over 65
391
472
(Source: Bureau of Labor Statistics, 4th quarter of 1999 data)
Let W(a) represent the median weekly income (in dollars) of women and M(a) represent the median
weekly income (in dollars) of men, both at age a years. Here are the quadratic regression equations for W
and M:
W(a) = -0.30a2 + 27.61a - 97.01
M(a) = -0.55a2 + 52.01a - 456.46
According to these models, at what age(s) is the median weekly income of men equal to the median
income of women?
8) The following table shows the median number of hours of leisure time per week for Americans in various
years.
Year Median Number of Leisure Hours per Week
1973
26.2
1980
19.2
1987
16.6
1993
18.8
1997
19.5
Let f(t) be the median number of hours of leisure time at t years since 1973. Use a graphing calculator to
plot a scatter diagram to describe the data. Using the same viewing window, draw the graph of the
quadratic model f(t) = 0.04t2 - 1.21t + 26.03. Use the model to estimate when the median number of hours
of leisure time was the lowest.
2
Answer Key
Testname: QUADRATIC MODELS
1) 40 units
2) 44 ft
3) 5.9 sec
4) vertex (- 1, 9);
x-int: (-4,0) and (2, 0);
y-int: (0, 8)
y
10
5
-10
-5
5
10
x
-5
-10
5) R(x) = -0.091x2 + 5.95x + 6337
6) M(x) = 0.04x2 - 1.21x + 26.03; 1988
7) 18, 80
8)
Median Number of Leisure Hours
28
f(t)
24
20
16
12
8
4
4
8
12
16
20
24
t
Years since 1973
The median number of hours of leisure time was the lowest in 1988.
3