1995 to 1980: about -19.239 Interpretation: See solution
C
c See solution
Practice makes perfect
a To graph the function, we will begin by changing the viewing window so that we can see the wanted graph properly. To do that we press WINDOW and change the variables using the given information.
Next, we will press Y= and write the function in one of the rows. Having written the function, we can press GRAPH to draw it. Then we can interpret the graph.
In this situation, t=0 corresponds to the year 1980. With this we can conclude that from 1980 to 2007, the number of theaters decreased. However, the decreases slowed down after 1995.
b The years 1980, 1995, and 2007 correspond to t=0, t=15, and t=27, respectively. Therefore, the average rates of change can be found as shown below.
From 1980 to 1995
r_(1980-1995)=d(15)-d(0)/15-0
From 1995 to 2007
r_(1995-2007)=d(27)-d(15)/27-15
To find d(0), d(15), and d(27), we will substitute t=0, t=15, and t=27 in the given function.
t
-0.141x^3+9.64x^2-232.5x+2421
d(t)
0
-0.141( 0)^3+9.64( 0)^2-232.5( 0)+2421
2421
15
-0.141( 15)^3+9.64( 15)^2-232.5( 15)+2421
626.625
27
-0.141( 27)^3+9.64( 27)^2-232.5( 27)+2421
395.757
Using the results, let's first find the average rate of change from 1980 to 1995.
The average rate of change from 1980 to 1995 is about -119.625 theaters per year. Proceeding in the same way, we can find the average rate of change from 1995 to 2007.
Years
Substitution
r
1980-1995
r_(1980-1995)=626.625-2421/15-0
-119.625
1995-2007
r_(1995-2007)=395.757-626.625/27-15
-19.239
The results verify our interpretation in Part A. From 1980 to 1995, about 120 theaters were closed in each year. However, from 1995 to 2007, about 20 theaters were closed.
c Let's first determine the end behavior of the function. The degree of the function is odd and its leading coefficient is negative. With this we can determine the end behavior of the function as shown below.
d(t)→ ∞ as t → - ∞
d(t)→ - ∞ as t → ∞
The model may be valid for the years before 1980. However, unlimited growth is not reasonable. The model can be used a few years later 2007 but the number of theaters cannot be negative. Therefore, the model should not be used in the long run.
