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 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]82 \%[/tex] compounded yearly, how much will be saved at the end of six years? Use the formula [tex]A=\frac{P[(1+\frac{r}{n})^{nt}-1]}{(\frac{r}{n})}[/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 (1)
Calculate the total annual expense: 0.98 × 30000 = 29400 .
Use the future value of annuity formula: A = ( n r ) P [( 1 + n r ) n t − 1 ] .
Substitute the values: P = 29400 , r = 0.82 , n = 1 , t = 6 .
Calculate the amount saved: A ≈ 1267199 .
Explanation
Understanding the Problem We are given the average operating and ownership costs per mile for a certain car, as well as the number of miles driven per year. We need to calculate the total annual expense for the car and the amount saved at the end of six years if the total annual expense is deposited into an IRA with a given interest rate.
Calculating Total Annual Expense First, we need to calculate the total annual expense for the car. The total cost per mile is $0.98 , and the number of miles driven per year is 30,000. So, the total annual expense is: 0.98 × 30000 = 29400
Understanding the Future Value Formula Next, we need to calculate the amount saved at the end of six years if the total annual expense is deposited into an IRA. 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 at the end of six years
P is the total annual expense (which we calculated in the previous step)
r is the interest rate
n is the number of times interest is compounded per year
t is the number of years
Identifying the Values In this case, we have:
P = 29400
r = 0.82 (82% expressed as a decimal)
n = 1 (compounded yearly)
t = 6
Substituting the Values into the Formula Substituting these values into the formula, we get: A = ( 1 0.82 ) 29400 [( 1 + 1 0.82 ) 1 × 6 − 1 ]
A = 0.82 29400 [( 1 + 0.82 ) 6 − 1 ]
A = 0.82 29400 [( 1.82 ) 6 − 1 ]
Calculating (1.82)^6 Now, we calculate ( 1.82 ) 6 :
( 1.82 ) 6 ≈ 36.3436
Calculating the Final Amount Substituting this back into the formula, we get: A = 0.82 29400 [ 36.3436 − 1 ]
A = 0.82 29400 [ 35.3436 ]
A = 0.82 1039101.84
A ≈ 1267197.37
Final Answer Rounding to the nearest dollar, the amount saved at the end of six years is $1 , 267 , 199 .
Examples
Understanding compound interest is crucial for long-term financial planning. For instance, if you invest a certain amount of money each year into a retirement account, the compound interest helps your investment grow significantly over time. This calculation is also useful when planning for college savings or any other long-term financial goals where consistent contributions and interest accumulation play a key role. By understanding these concepts, you can make informed decisions about your financial future and plan effectively for your goals.
