Suppose that a certain car has the following average operating and ownership costs:
| | | |
| :--------- | :--------- | :---- |
| | | |
| Operating | Ownership | Total |
| [tex]$0.26[/tex] | [tex]$0.72[/tex] | [tex]$0.98[/tex] |
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 six 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]
Answer (2)
Calculate the total annual expense by multiplying the total cost per mile by the number of miles driven per year: 0.98 × 30000 = $29 , 400 .
Use the future value of an annuity formula to calculate the amount saved after 6 years: A = ( n r ) P [( 1 + n r ) n t − 1 ] .
Substitute the given values into the formula: A = ( 1 0.082 ) 29400 [( 1 + 1 0.082 ) 1 × 6 − 1 ] .
Calculate the amount saved: A ≈ $216 , 779 . The total annual expense is $29 , 400 and the amount saved after 6 years is $216 , 779 .
Explanation
Understanding the Problem We are given the average operating and ownership costs per mile for a car, and we need to calculate the total annual expense and the amount saved after six years with annual deposits into an IRA.
Calculating Total Annual Expense First, we need to calculate the total annual expense. The total cost per mile is $0.98, and the car is driven 30,000 miles per year. To find the total annual expense, we multiply these two values: T o t a l \[ 0.2 c m ] A nn u a l \[ 0.2 c m ] E x p e n se = T o t a l \[ 0.2 c m ] C os t \[ 0.2 c m ] p er \[ 0.2 c m ] M i l e × M i l es \[ 0.2 c m ] Dr i v e n \[ 0.2 c m ] p er \[ 0.2 c m ] Y e a r
Total Annual Expense Calculation Substituting the given values, we have: T o t a l \[ 0.2 c m ] A nn u a l \[ 0.2 c m ] E x p e n se = $0.98 × 30 , 000 = $29 , 400
Understanding the Future Value Formula Next, we need to calculate the amount saved at the end of six 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 is the amount saved - P is the annual deposit (total annual expense) - r is the interest rate - n is the number of times the interest is compounded per year - t is the number of years
Substituting Values into the Formula In this case, we have: - P = $29 , 400 - r = 0.082 - n = 1 - t = 6 Substituting these values into the formula, we get: A = ( 1 0.082 ) 29400 [( 1 + 1 0.082 ) 1 × 6 − 1 ]
Calculating the Future Value First, calculate ( 1 + 0.082 ) 6 : ( 1 + 0.082 ) 6 = ( 1.082 ) 6 ≈ 1.604588 Now, substitute this back into the formula: A = 0.082 29400 [ 1.604588 − 1 ] = 0.082 29400 [ 0.604588 ]
Final Calculation and Rounding A = 0.082 17775.8832 ≈ 216779.06 Rounding to the nearest dollar, we get: A ≈ $216 , 779
Final Answer Therefore, if you drive 30,000 miles per year, the total annual expense for this car is $29,400. 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 six years will be approximately $216,779.
Examples
Understanding the costs associated with owning and operating a vehicle is crucial for budgeting and financial planning. This problem demonstrates how to calculate the total annual expense of a car and how to project the future value of annual savings deposited into an investment account. For example, knowing these costs can help you decide whether to lease or buy a car, or whether it's more economical to use public transportation. Furthermore, understanding the power of compound interest can motivate you to start saving early for long-term goals like retirement.
The total annual expense for driving the car for 30,000 miles is $29,400. If this amount is deposited in an IRA at an 8.2% annual interest rate compounded yearly for six years, approximately $216,779 will be saved. This demonstrates the importance of understanding both vehicle costs and the long-term benefits of saving with compound interest.
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