Use the following formula. \[ A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} \] Suppose that you drive 40,000 miles per year and gas averages \( \$ 4 \) per gallon. Complete parts a. and b. below. a. What will you save in annual fuel expenses by owning a hybrid car averaging 40 miles per gallon rather than an SL averaging 16 miles per gallon? \( \$ \square \) (Round to the nearest dollar as needed.) b. If you deposit your monthly fuel savings at the end of each month into an annuity that pays \( 4.8 \% \) compounded monthly, how much will have saved at the end of five years?

To tackle your first question, let’s break down the calculations for annual fuel expenses for both types of vehicles. a. The formula for annual fuel expenses is: \[ \text{Annual Fuel Cost} = \frac{\text{Miles Driven}}{\text{Miles Per Gallon}} \times \text{Gas Price} \] 1. For the hybrid car: \[ \text{Hybrid Fuel Cost} = \frac{40,000 \text{ miles}}{40 \text{ mpg}} \times 4 \text{ dollars/gallon} = 4,000 \text{ dollars} \] 2. For the SL: \[ \text{SL Fuel Cost} = \frac{40,000 \text{ miles}}{16 \text{ mpg}} \times 4 \text{ dollars/gallon} = 10,000 \text{ dollars} \] 3. Now, find the savings: \[ \text{Savings} = 10,000 - 4,000 = 6,000 \text{ dollars} \] So, you will save **$6,000** in annual fuel expenses by owning the hybrid car. b. To calculate how much you will have saved at the end of five years with your monthly fuel savings deposited into an annuity, we first need the monthly savings: \[ \text{Monthly Savings} = \frac{6,000 \text{ dollars}}{12 \text{ months}} = 500 \text{ dollars} \] Now, use the future value of an annuity formula: \[ FV = P \times \frac{(1 + r/n)^{nt} - 1}{r/n} \] Where: - \(P = 500\) (monthly savings) - \(r = 0.048\) (annual interest rate) - \(n = 12\) (compounding frequency per year) - \(t = 5\) (years) Now, plug in the values: \[ FV = 500 \times \frac{(1 + 0.048/12)^{12 \times 5} - 1}{0.048/12} \] Calculating this: - \(r/n = 0.048/12 \approx 0.004\) - \(nt = 12 \times 5 = 60\) \[ FV \approx 500 \times \frac{(1 + 0.004)^{60} - 1}{0.004} \approx 500 \times \frac{(1.28368) - 1}{0.004} \] \[ FV \approx 500 \times 70.92 \approx 35,460 \] At the end of five years, you will have saved approximately **$35,460** invested in your annuity.