The dollar value \( v(t) \) of a certain car model that is \( t \) years old is given by the following exponential function. \[ v(t)=24,500(0.84)^{t} \] Find the value of the car after 4 years and after 10 years. Round your answers to the nearest dollar as necessary. value after 4 years: value after 10 years:

To determine the value of the car after 4 years and after 10 years using the exponential depreciation function: \[ v(t) = 24,500 \times (0.84)^t \] ### **1. Value After 4 Years (\( t = 4 \))** \[ v(4) = 24,500 \times (0.84)^4 \] First, calculate \( (0.84)^4 \): \[ (0.84)^4 \approx 0.4979 \] Then, multiply by the initial value: \[ v(4) = 24,500 \times 0.4979 \approx 12,298 \text{ dollars} \] ### **2. Value After 10 Years (\( t = 10 \))** \[ v(10) = 24,500 \times (0.84)^{10} \] First, calculate \( (0.84)^{10} \): \[ (0.84)^{10} \approx 0.1751 \] Then, multiply by the initial value: \[ v(10) = 24,500 \times 0.1751 \approx 4,291 \text{ dollars} \] ### **Final Answers** - **Value after 4 years:** \$12,298 - **Value after 10 years:** \$4,291