### Part (a)
To write the equation of the exponential function that models the retail price of the automobile [tex]\( t \)[/tex] years after 1999, we begin with the general form of the exponential function:
[tex]\[ y = a(1 + r)^t \][/tex]
Where:
- [tex]\( y \)[/tex] represents the retail price of the automobile after [tex]\( t \)[/tex] years.
- [tex]\( a \)[/tex] is the initial retail price of the automobile.
- [tex]\( r \)[/tex] is the growth rate per year.
- [tex]\( t \)[/tex] is the number of years after 1999.
Given the information:
- The initial price [tex]\( a \)[/tex] in 1999 is \[tex]$28,000.
- The growth rate \( r \) is 2% per year, or 0.02 in decimal form.
Thus, the equation that models the retail price of the automobile \( t \) years after 1999 is:
\[ P(t) = 28000(1.02)^t \]
So the correct function is:
\[ P(t) = 28,000 \cdot (1.02)^t \]
### Part (b)
Now, we use the model to predict the retail price of the automobile in 2014.
First, determine the number of years \( t \) between 1999 and 2014:
\[ t = 2014 - 1999 = 15 \]
Next, substitute \( t = 15 \) into the model:
\[ P(15) = 28000 \cdot (1.02)^{15} \]
Using the model to find the retail price:
\[ P(15) \approx 28000 \cdot 1.345852 \]
Therefore, the retail price of the automobile in 2014 is approximately:
\[ P(15) \approx 37684.31 \]
Rounding to the nearest dollar:
\[ \boxed{37684} \]
So, according to the model, the retail price of the automobile in 2014 is approximately \$[/tex]37,684.
