6.1.2 Exam: Semester 2 Exam Question 14 of 40 Prakash bought a new car at the dealership for \( \$ 27,000 \). It is estimated that the value of the car will decrease \( 7 \% \) each year. Which exponential function models the value \( v \) of the car after \( t \) years? A. \( v=27,000(0.3)^{t} \) B. \( v=27,000(1.3)^{t} \) c. \( v=27,000(1.03)^{t} \) D. \( v=27,000(0.93)^{t} \)

To model the depreciation of Prakash's new car, we use an exponential decay function. Here's how to determine the correct function: 1. **Initial Value (\( v_0 \))**: The car's initial value is \$27,000. 2. **Depreciation Rate**: The car loses 7% of its value each year. This means it retains \( 100\% - 7\% = 93\% \) of its value annually. 3. **Exponential Decay Formula**: The general formula for exponential decay is: \[ v = v_0 \times (1 - r)^t \] where: - \( v \) is the value after \( t \) years, - \( v_0 \) is the initial value, - \( r \) is the depreciation rate. Applying the values: \[ v = 27,000 \times (0.93)^t \] **Option D** correctly represents this formula: \[ \boxed{D.\ v=27,000(0.93)^{t}} \]