Exponential and Logarithmic Functions
Evaluating an exponential function that models a real-world situation

The dollar value $v(t)$ of a certain car model that is $t$ years old is given by the following exponential function.
\[
v(t)=20,000(0.90)^{t}
\]

Find the value of the car after 7 years and after 11 years.
Round your answers to the nearest dollar as necessary.
Value after 7 years:
Value after 11 years:

Step 1 ：The problem is asking for the value of the car after 7 years and 11 years. This can be calculated by substituting the values of 7 and 11 for t in the given function \(v(t)=20000(0.90)^t\).

Step 2 ：Substitute 7 for t in the function to find the value of the car after 7 years: \(v(7)=20000(0.90)^7\).

Step 3 ：Calculate the value to get \(v(7) = 9566\).

Step 4 ：Substitute 11 for t in the function to find the value of the car after 11 years: \(v(11)=20000(0.90)^{11}\).

Step 5 ：Calculate the value to get \(v(11) = 6276\).

Step 6 ：Final Answer: The value of the car after 7 years is \(\boxed{9566}\) dollars and after 11 years is \(\boxed{6276}\) dollars.