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]$A=\frac{\left[\left(\frac{r}{n}\right)-1\right]}{\left(\frac{r}{n}\right)}$[/tex].
[tex]$P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]$[/tex]
[tex]$\left(\frac{r}{n}\right)$[/tex]
a. If you drive 30,000 miles per year, the total annual expense for this car is $ $\square$
(Round to the nearest dollar as needed.)
Answer (2)
Calculate the total annual expense by multiplying the total cost per mile by the annual mileage: T o t a l A nn u a l E x p e n se = $0.96 × 30 , 000 = $28 , 800 .
Use the future value of an annuity formula to calculate the amount saved at the end of seven years: A = P r [ ( 1 + r ) t − 1 ] , where P = $28 , 800 , r = 0.082 , and t = 7 .
Substitute the values into the formula: A = 28800 0.082 [ ( 1 + 0.082 ) 7 − 1 ] .
Calculate the future value and round to the nearest dollar: A = 258555 .
Explanation
Understanding the Problem Let's break down this problem step by step so we can understand how to calculate the total expenses for the car and how much money will be saved in the IRA account.
Calculating Total Annual Expense First, we need to calculate the total annual expense for the car. We know the average total cost per mile is $0.96, and you drive 30,000 miles per year. To find the total annual expense, we multiply these two values.
Total Annual Expense Calculation 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 T o t a l A nn u a l E x p e n se = $28 , 800
Understanding the Future Value Formula Next, we need to determine how much will be saved at the end of seven years if the total annual expense is deposited into an IRA with an interest rate of 8.2% compounded yearly. We will use the future value of an annuity formula: A = P n r [ ( 1 + n r ) n t − 1 ] Where:
A is the future value of the annuity
P is the annual deposit (total annual expense)
r is the interest rate (as a decimal)
n is the number of times the interest is compounded per year
t is the number of years
Calculating the Future Value In this case:
P = $28 , 800
r = 0.082
n = 1
t = 7 Plugging these values into the formula, we get: A = 28800 0.082 [ ( 1 + 0.082 ) 1 × 7 − 1 ] A = 28800 0.082 [ ( 1.082 ) 7 − 1 ] A = 28800 0.082 [ 1.7361643145023198 − 1 ] A = 28800 0.082 0.7361643145023198 A = 28800 × 8.977613591491705 A = 258555.2714349611
Final Answer Rounding to the nearest dollar, the amount saved at the end of seven years will be $258,555.
Examples
Understanding the costs associated with owning a car, like operating and ownership expenses, is crucial for budgeting and financial planning. By calculating these costs and planning to save the equivalent amount in an investment account like an IRA, you can secure your financial future. This approach is similar to saving for retirement or a down payment on a house, where consistent contributions over time can lead to significant financial growth due to the power of compound interest. For example, knowing your car expenses allows you to plan for future car purchases or other significant investments.
The total annual expense for driving 30,000 miles per year is $28,800. If this amount is saved in an IRA at 8.2% interest compounded yearly, it will grow to approximately $258,576 after seven years.
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