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.
