Use the following formula. $$ \mathrm{A}=\frac{\mathrm{P}\left[\left(1+\frac{r}{n}\right)^{\mathrm{nt}}-1\right]}{\left(\frac{r}{n}\right)} $$ Suppose that you drive 50,000 miles per year and gas averages $$ \$ 4 $$ per gallon. Complete parts a. and $$ \mathbf{b} $$. below. a. What will you save in annual fuel expenses by owning a hybrid car averaging 50 miles per gallon rather than an SUV averaging 25 miles per gallon? $$ \$ 4000 $$ (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 $$ 5.2 \% $$ compounded monthly, how much will have saved at the end of five years?

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Let's break down the problem step by step.

### Part a: Annual Fuel Expenses Savings

1. **Calculate the annual fuel expenses for the SUV:**
   - Miles driven per year: $$ 50,000 $$
   - Fuel efficiency of the SUV: $$ 25 $$ miles per gallon
   - Cost of gas: $$ \$ 4 $$ per gallon

First, we need to find out how many gallons of gas the SUV uses in a year:
   $$
   \text{Gallons used by SUV} = \frac{\text{Miles driven}}{\text{Fuel efficiency}} = \frac{50000}{25} = 2000 \text{ gallons}
   $$

Now, calculate the total cost for the SUV:
   $$
   \text{Cost for SUV} = \text{Gallons used} \times \text{Cost per gallon} = 2000 \times 4 = 8000
   $$

2. **Calculate the annual fuel expenses for the hybrid car:**
   - Fuel efficiency of the hybrid: $$ 50 $$ miles per gallon

Now, find out how many gallons of gas the hybrid uses in a year:
   $$
   \text{Gallons used by hybrid} = \frac{50000}{50} = 1000 \text{ gallons}
   $$

Calculate the total cost for the hybrid:
   $$
   \text{Cost for hybrid} = 1000 \times 4 = 4000
   $$

3. **Calculate the savings:**
   $$
   \text{Savings} = \text{Cost for SUV} - \text{Cost for hybrid} = 8000 - 4000 = 4000
   $$

Thus, the savings in annual fuel expenses by owning a hybrid car rather than an SUV is $$ \$ 4000 $$.

### Part b: Future Value of Monthly Savings in an Annuity

1. **Monthly savings:**
   Since the annual savings is $$ \$ 4000 $$, the monthly savings will be:
   $$
   \text{Monthly savings} = \frac{4000}{12} \approx 333.33
   $$

2. **Annuity formula:**
   We will use the future value of an annuity formula:
   $$
   \mathrm{A} = \frac{\mathrm{P}\left[\left(1+\frac{r}{n}\right)^{\mathrm{nt}}-1\right]}{\left(\frac{r}{n}\right)}
   $$
   where:
   - $$ P = 333.33 $$ (monthly savings)
   - $$ r = 0.052 $$ (annual interest rate)
   - $$ n = 12 $$ (number of compounding periods per year)
   - $$ t = 5 $$ (number of years)

3. **Substituting values into the formula:**
   - First, calculate $$ \frac{r}{n} $$:
   $$
   \frac{r}{n} = \frac{0.052}{12} \approx 0.00433333
   $$
   - Now calculate $$ nt $$:
   $$
   nt = 12 \times 5 = 60
   $$
   - Substitute into the formula:
   $$
   \mathrm{A} = \frac{333.33\left[\left(1+0.00433333\right)^{60}-1\right]}{0.00433333}
   $$

Now, let's calculate this value.

I'll perform the calculation for the future value of the annuity.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+0.00433333\right)^{60}$$
- step1: Add the numbers:
  $$1.00433333^{60}$$
- step2: Convert the expressions:
  $$\left(\frac{100433333}{100000000}\right)^{60}$$
- step3: Use the properties of exponents:
  $$\frac{100433333^{60}}{100000000^{60}}$$
Calculate or simplify the expression $$ 333.33*((1.296202)-1)/0.00433333 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{333.33\left(1.296202-1\right)}{0.00433333}$$
- step1: Subtract the numbers:
  $$\frac{333.33\times 0.296202}{0.00433333}$$
- step2: Multiply the numbers:
  $$\frac{98.73301266}{0.00433333}$$
- step3: Convert the expressions:
  $$\frac{\frac{4936650633}{50000000}}{\frac{433333}{100000000}}$$
- step4: Multiply by the reciprocal:
  $$\frac{4936650633}{50000000}\times \frac{100000000}{433333}$$
- step5: Reduce the numbers:
  $$4936650633\times \frac{2}{433333}$$
- step6: Multiply:
  $$\frac{4936650633\times 2}{433333}$$
- step7: Multiply:
  $$\frac{9873301266}{433333}$$
Now that we have the necessary calculations, let's summarize the results for part b.

### Part b: Future Value of Monthly Savings in an Annuity

1. **Future Value Calculation:**
   We found that:
   $$
   (1 + 0.00433333)^{60} \approx 1.296202
   $$
   Substituting this into the annuity formula:
   $$
   \mathrm{A} = \frac{333.33 \left[1.296202 - 1\right]}{0.00433333}
   $$
   This simplifies to:
   $$
   \mathrm{A} \approx \frac{333.33 \times 0.296202}{0.00433333} \approx 22784.56
   $$

Thus, at the end of five years, the total amount saved in the annuity will be approximately $$ \$ 22,785 $$ (rounded to the nearest dollar).

### Final Answers:
- a. Annual fuel expenses savings: $$ \$ 4000 $$
- b. Total savings at the end of five years: $$ \$ 22,785 $$
a. You save \$4,000 annually by owning a hybrid car instead of an SUV. b. After five years, your total savings will be approximately \$22,785.

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