Suppose that a certain car has the following average operating and ownership costs.
| | | |
| :------------ | :------------ | :---- |
| | | Total |
| Operating | Ownership | |
| \$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 \$28800.
(Round to the nearest dollar as needed.)
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, the amount saved at the end of seven years is \$ [ ]
(Round to the nearest dollar as needed.)
Answer (1)
Calculate the total annual expense: $0.96 \t\times 30000 = $28800.
Substitute the values into the formula: A = ( 1 0.082 ) 28800 [ ( 1 + 1 0.082 ) 1 × 7 − 1 ] .
Simplify the expression: A = 0.082 28800 × ( 1.08 2 7 − 1 ) ≈ 258530.75 .
Round the final amount to the nearest dollar: The amount saved at the end of seven years is 258531 .
Explanation
Understanding the Problem We are given the average costs per mile for a certain car and asked to calculate the total annual expense if we drive 30,000 miles per year. We are also asked to calculate the amount saved at the end of seven years if the total annual expense is deposited at the end of each year into an IRA paying 8.2% compounded yearly.
Calculating Total Annual Expense First, we need to calculate the total annual expense. We are given that the total cost per mile is $0.96 and we drive 30,000 miles per year. Therefore, the total annual expense is: 0.96 × 30000 = 28800
Understanding the Formula Next, we need to calculate the amount saved at the end of seven years. We are given the formula: A = ( n r ) P [ ( 1 + n r ) n t − 1 ] where:
A is the amount saved at the end of seven years
P is the total annual expense ($28800)
r is the interest rate (8.2% or 0.082)
n is the number of times interest is compounded per year (1)
t is the number of years (7)
Calculating the Amount Saved Now, we substitute the values into the formula: 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 [ 1.7361643145 − 1 ] A = 0.082 28800 [ 0.7361643145 ] A = 0.082 21199.52124656 A = 258530.746909
Final Answer Rounding to the nearest dollar, the amount saved at the end of seven years is $258,531.
Examples
Understanding compound interest is crucial for long-term financial planning. For instance, saving regularly for retirement involves depositing a certain amount periodically into an investment account that earns interest. The formula used in this problem helps calculate the future value of these investments, allowing individuals to estimate their savings over time and make informed decisions about their financial goals. This concept is also applicable in other scenarios, such as calculating loan payments or the growth of a business investment.
