The dollar value \( v(t) \) of a certain car model that is \( t \) years old is given by the following exponential function. \[ v(t)=27,500(0.80)^{t} \] Find the value of the car after 7 years and after 13 years. Round your answers to the nearest dollar as necessary.

To find the value of the car after \( t = 7 \) years, you can substitute \( t = 7 \) into the function: \[ v(7) = 27,500(0.80)^{7} \approx 27,500 \times 0.2097152 \approx 5,764 \] So, after 7 years, the value of the car is approximately $5,764. Now for \( t = 13 \) years: \[ v(13) = 27,500(0.80)^{13} \approx 27,500 \times 0.05764801 \approx 1,587 \] Thus, after 13 years, the value of the car is approximately $1,587.