Suppose that a certain car has the following average operating and ownership costs.
| | | |
| :--------- | :--------- | :---- |
| | Ownership | Total |
| Operating | | |
| $ 0.24 | $ 0.72 | $ 0.96 |
a. If you drive 30,000 miles per year, what is the total annual expense for this car?
b. If the total annual expense for this car is deposited at the end of each year into an IRA paying [tex]$8.2 \%$[/tex] compounded yearly, how much will be saved at the end of seven years? Use the formula [tex]$A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)}$[/tex].
a. If you drive 30,000 miles per year, the total annual expense for this car is $ (Round to the nearest dollar as needed.)
Answer (2)
Calculate the total annual expense by multiplying the cost per mile by the annual mileage: 0.96 × 30000 = $28800 .
Use the future value of annuity formula to calculate the amount saved after 7 years: A = ( n r ) P [( 1 + n r ) n t − 1 ] .
Substitute the given values into the formula: A = ( 1 0.082 ) 28800 [( 1 + 1 0.082 ) 1 × 7 − 1 ] .
Calculate the final amount: A ≈ 258531 .
Explanation
Problem Analysis We are given the average operating and ownership costs per mile for a certain car. We need to calculate the total annual expense for driving 30,000 miles per year and then determine how much would be saved after seven years if the total annual expense is deposited into an IRA with an interest rate of 8.2% compounded yearly.
Calculating Total Annual Expense First, we need to calculate the total annual expense. The total cost per mile is given as $0.96. If you drive 30,000 miles per year, the total annual expense is calculated as follows: T o t a l A nn u a l E x p e n se = T o t a l C os t p er M i l e × A nn u a l M i l e a g e T o t a l A nn u a l E x p e n se = $0.96 × 30 , 000 = $28 , 800
Understanding the Future Value Formula Next, we need to calculate the amount saved at the end of seven years. We are given the formula for the future value of an annuity: A = ( n r ) P [ ( 1 + n r ) n t − 1 ] Where:
A = the amount of money accumulated after n years, including interest.
P = the principal amount (the initial amount of money).
r = the annual interest rate (as a decimal).
n = the number of times that interest is compounded per year.
t = the number of years the money is invested or borrowed for
Calculating the Amount Saved In this case:
P = $28,800 (the total annual expense)
r = 0.082 (8.2% annual interest rate)
n = 1 (compounded yearly)
t = 7 (number of years) Substituting these values into the formula, we get: A = ( 1 0.082 ) 28800 [ ( 1 + 1 0.082 ) 1 × 7 − 1 ] A = 0.082 28800 [ ( 1 + 0.082 ) 7 − 1 ] A = 0.082 28800 [ ( 1.082 ) 7 − 1 ] A = 0.082 28800 [ 1.736164 − 1 ] A = 0.082 28800 [ 0.736164 ] A = 0.082 21199.5072 A = 258530.5756 Rounding to the nearest dollar, we get $258,531.
Final Answer Therefore, if you drive 30,000 miles per year, the total annual expense for this car is $28,800. If this amount is deposited at the end of each year into an IRA paying 8.2% compounded yearly, the amount saved at the end of seven years will be approximately $258,531.
Examples
Understanding the costs associated with owning and operating a car is crucial for budgeting and financial planning. The concepts of annual expenses and compound interest are applicable in many real-life scenarios. For example, when planning for retirement, individuals need to estimate their annual expenses and calculate how much they need to save each year to reach their financial goals. The formula for the future value of an annuity helps in determining the amount saved over time, considering the power of compound interest. This knowledge is also useful when making investment decisions or evaluating loan options.
The total annual expense for driving 30,000 miles per year is $28,800. If this amount is deposited annually into an IRA at an 8.2% interest rate compounded yearly, the savings at the end of seven years will be approximately $258,531.
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