Question

From 1980 to 2007 the number of drive in theaters in the United States can be modeled by the function

d(t) = - 0.141t³ + 9.64t² - 232.5t + 2421
where d(t) is the number of open theaters and t is the number of years after 1980.

a. Use a graphing calculator to graph the function for the interval 0 ≤ t ≤ 27. Describe the behavior of the graph on this interval.
b. What is the average rate of change in the number of drive-in-movie theaters from 1980 to 1995 and from 1995 - 2007? Interpret the average rates of change.
c. Do you think this model can be used for years before 1980 or after 2007? Explain

Answer 1

Explanation:

See the attached graph.

a. The graph is attached. The slope is very negative and begins to level out in future years before starting to turn down again.

b. The average rate of change is the slope of the line connecting the two points at 1980 and 1995 (x = 0 and 15). A calculator is used to compute the values of y (theaters) for these 2 values of x:

(0,2421) and (15,626.62)

Rise = -1794.4

Run = 15

Slope = -119.6

For the years between 1995 and 2007:

(15,626.6) and (27,395.76)

Rise = (395.76 - 626.6) = -230.87 Rate of change for 1980 to 1995

Run = (27 - 15) = 12

Slope = -230.87/12 = -19.24 Rate of change for 1995 to 2007

c. No. The model may not be useful either before 1980 nor after 2007. The curve would angle sharply upward before 1980. This is unlikely to represent the actual situation. The number of theaters decreases after 2007 according to the formula. This not take into account any social or technology trends that may impact decisions on whether to go to a theater.