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Suppose that you drive 30,000 miles per year and gas averages [tex]$4 per gallon. Complete parts a and b below.

a. What will you save in annual fuel expenses by owning a hybrid car averaging 30 miles per gallon rather than an SUV averaging 9 miles per gallon?
$[/tex]_____
(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 you have saved at the end of six years?
$_____
(Round to the nearest dollar as needed.)

Sagot :

Let's solve this step-by-step.

### Part a: Calculate annual fuel expenses savings

1. Miles driven per year: [tex]\( 30,000 \)[/tex] miles.
2. Gas price per gallon: [tex]\( \$4 \)[/tex] per gallon.
3. Fuel efficiency of hybrid car: [tex]\( 30 \)[/tex] miles per gallon.
4. Fuel efficiency of SUV: [tex]\( 9 \)[/tex] miles per gallon.

#### Hybrid Car:
- Gallons of gas used per year:
[tex]\[ \text{Gallons used (Hybrid)} = \frac{30,000 \text{ miles}}{30 \text{ mpg}} = 1,000 \text{ gallons} \][/tex]
- Annual fuel cost for hybrid car:
[tex]\[ \text{Annual fuel cost (Hybrid)} = 1,000 \text{ gallons} \times \$4/\text{gallon} = \$4,000 \][/tex]

#### SUV:
- Gallons of gas used per year:
[tex]\[ \text{Gallons used (SUV)} = \frac{30,000 \text{ miles}}{9 \text{ mpg}} \approx 3,333.33 \text{ gallons} \][/tex]
- Annual fuel cost for SUV:
[tex]\[ \text{Annual fuel cost (SUV)} \approx 3,333.33 \text{ gallons} \times \$4/\text{gallon} = \$13,333.33 \][/tex]

#### Annual Savings:
- Annual savings by owning a hybrid:
[tex]\[ \text{Annual Savings} = \$13,333.33 - \$4,000 = \$9,333.33 \][/tex]

So, the savings in annual fuel expenses is approximately \[tex]$9,333. ### Part b: Calculate future value of annuity To calculate the future value of the savings deposited into an annuity: 1. Monthly savings: \[ P = \frac{\$[/tex]9,333.33}{12} \approx \[tex]$777.78 \] 2. Interest rate (annual): \[ r = 4.8\% = 0.048 \] 3. Number of compounding periods per year: \[ n = 12 \] 4. Total time (years): \[ t = 6 \] Using the Future Value of Annuity formula: \[ A = \frac{P \left( \left( 1 + \frac{r}{n} \right)^{nt} - 1 \right)}{\frac{r}{n}} \] Substituting in the values: \[ A = \frac{777.78 \left( \left( 1 + \frac{0.048}{12} \right)^{12 \times 6} - 1 \right)}{\frac{0.048}{12}} \] Calculating the future value of the annuity, we get approximately: \[ A \approx \$[/tex]64,748
\]

So, the amount saved at the end of six years is approximately \$64,748.
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