Use the following formula.

Question

$A=\frac {P[(1+\frac {r}{n})^{nt}-1]}{(\frac {r}{n})}$
Suppose that you drive 50,000 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 50 miles per gallon rather than an SUV
averaging 25 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 $5.2\% $
compounded monthly, how much will have saved at the end of five years?
$\$ \square $
(Round to the nearest dollar as needed

Answer 1

To solve this problem, we will break it down into the two parts (a and b) as specified.

Part a: Annual fuel expenses savings

1. Calculate the amount of gas used for each type of vehicle:

   • For the hybrid car (50 miles per gallon): \[ \text{Gallons used} = \frac{\text{Total miles}}{\text{Miles per gallon}} = \frac{50000}{50} = 1000 \text{ gallons} \]

   • For the SUV (25 miles per gallon): \[ \text{Gallons used} = \frac{50000}{25} = 2000 \text{ gallons} \]

2. Calculate the annual fuel costs for each type of vehicle:

   • For the hybrid car: \[ \text{Cost} = \text{Gallons used} \times \text{Price per gallon} = 1000 \times 4 = 4000 \text{ dollars} \]

   • For the SUV: \[ \text{Cost} = 2000 \times 4 = 8000 \text{ dollars} \]

3. Calculate the savings: \[ \text{Savings} = \text{Cost of SUV} - \text{Cost of Hybrid} = 8000 - 4000 = 4000 \text{ dollars} \]

So, the answer for part a is: \[ \boxed{4000} \]

Part b: Future value of savings in an annuity

We are given that you will save $4000 per year, which translates to a monthly savings of: \[ \text{Monthly Savings} = \frac{4000}{12} \approx 333.33 \text{ dollars} \]

1. Key variables for the annuity calculation:

   • Monthly savings (\( P \)) = 333.33
   • Annual interest rate (\( r \)) = 5.2% = 0.052
   • Monthly interest rate (\( \frac{r}{n} \)) = \( \frac{0.052}{12} \approx 0.00433333 \)
   • Total number of deposits (months) (\( nt \)) = 5 years × 12 months/year = 60 months

2. Use the formula for the future value of an annuity: \[ A = \frac{P \left( \left(1 + \frac{r}{n}\right)^{nt} - 1 \right)}{\frac{r}{n}} \] Plugging in our values: \[ A = \frac{333.33 \left( \left(1 + 0.00433333\right)^{60} - 1 \right)}{0.00433333} \]

3. Calculate \( \left( 1 + 0.00433333 \right)^{60} \): \[ \left( 1 + 0.00433333 \right)^{60} \approx \left( 1.00433333 \right)^{60} \approx 1.28368 \]

4. Now substitute back into the equation: \[ A = \frac{333.33 \left( 1.28368 - 1 \right)}{0.00433333} = \frac{333.33 \times 0.28368}{0.00433333} \] \[ A = \frac{94.53}{0.00433333} \approx 21840.07 \]

Therefore, the rounded future value of the savings is: \[ \boxed{21840} \]