To predict the price of a movie ticket in year 20 using the given regression equation [tex]\( y = 6.94 \cdot (1.02^x) \)[/tex], follow these steps:
1. Identify the given parameters in the regression equation:
[tex]\[
y = 6.94 \cdot (1.02^x)
\][/tex]
Here, [tex]\( 6.94 \)[/tex] is the base price in 2007 (year 0), [tex]\( 1.02 \)[/tex] is the growth rate, and [tex]\( x \)[/tex] is the number of years since 2007.
2. Determine the value of [tex]\( x \)[/tex] for year 20:
[tex]\[
x = 20
\][/tex]
This represents the year 2027 because [tex]\( 2007 + 20 = 2027 \)[/tex].
3. Substitute [tex]\( x = 20 \)[/tex] into the regression equation:
[tex]\[
y = 6.94 \cdot (1.02^{20})
\][/tex]
4. Calculate [tex]\( (1.02^{20}) \)[/tex], which is approximately 1.485947:
[tex]\[
1.02^{20} \approx 1.485947
\][/tex]
5. Multiply this factor by the base price of the ticket:
[tex]\[
y = 6.94 \cdot 1.485947 \approx 10.312474928089783
\][/tex]
6. Round the result to the nearest cent:
[tex]\[
y \approx \$10.31
\][/tex]
Therefore, the best prediction of the price of a movie ticket in year 20 (2027) is [tex]\(\$10.31\)[/tex].
Hence, the correct answer is:
B. [tex]$\$[/tex] 10.31$
