Write a function in terms of \( t \) that represents the situation. A smartphone costs \( \$ 850 \), and its resell value decreases by \( 11.2 \% \) each year

To model the resell value of the smartphone over time, we can use an exponential decay function. Here's how to construct the function step-by-step: 1. **Initial Cost**: The initial cost of the smartphone is \$850. This serves as the starting value when \( t = 0 \). 2. **Depreciation Rate**: The resell value decreases by 11.2% each year. This means that each year, the smartphone retains \( 100\% - 11.2\% = 88.8\% \) of its value from the previous year. 3. **Exponential Decay Function**: The general form of an exponential decay function is: \[ V(t) = V_0 \times (1 - r)^t \] where: - \( V(t) \) is the value after \( t \) years, - \( V_0 \) is the initial value (\$850), - \( r \) is the depreciation rate (0.112). 4. **Substitute the Values**: \[ V(t) = 850 \times (1 - 0.112)^t \] \[ V(t) = 850 \times 0.888^t \] **Final Function**: \[ V(t) = 850 \times (0.888)^t \] **Example Usage**: - After 1 year: \[ V(1) = 850 \times 0.888^1 = 850 \times 0.888 = \$755 \] - After 2 years: \[ V(2) = 850 \times 0.888^2 \approx \$670.56 \] This function effectively models the decreasing resell value of the smartphone over time. \[ V(t) = 850 \times \left(0.888\right)^{\, t} \]