To solve this problem, we need to use the given exponential regression equation [tex]\( y = 9.79 \cdot (0.8213)^r \)[/tex], where [tex]\( y \)[/tex] represents the number of VHS movie rentals (in millions) and [tex]\( r \)[/tex] is the number of years since the year 2000.
We are asked to predict the number of VHS movie rentals in the year 2011. The steps are as follows:
1. First, determine the value of [tex]\( r \)[/tex]. Since [tex]\( r \)[/tex] is the number of years since 2000,
[tex]\[
r = 2011 - 2000 = 11.
\][/tex]
2. Substitute [tex]\( r = 11 \)[/tex] into the exponential regression equation:
[tex]\[
y = 9.79 \cdot (0.8213)^{11}.
\][/tex]
3. Compute [tex]\( (0.8213)^{11} \)[/tex]:
[tex]\[
(0.8213)^{11} \approx 0.1147.
\][/tex]
4. Multiply 9.79 by 0.1147 to find [tex]\( y \)[/tex]:
[tex]\[
y = 9.79 \cdot 0.1147 \approx 1.1228.
\][/tex]
So, the predicted number of VHS movie rentals in 2011 is approximately [tex]\( 1.1228 \)[/tex] million.
From the choices provided:
a. 0
b. 0.11
c. 88.45
d. 1.12
The closest and most appropriate answer is:
d. 1.12
