Question
When Martin was born, his father set up a $5,000 college fund for him with an annual compound interest rate of 7.3%.
What is the total value of this account after 18 years, and what was the interest earned on the account? Round the answers to two decimal places.
*Remember to use your STAAR Reference Sheet for support!
(2 points)
The final amount of the college fund is $
and the amount of interest earned on the account is $
Answers
GPT-4o mini
To calculate the total value of the college fund after 18 years with an initial amount of $5,000 and an annual compound interest rate of 7.3%, we can use the formula for compound interest:
\[
A = P(1 + r)^n
\]
where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of years the money is invested or borrowed for.
In this case:
- \(P = 5000\)
- \(r = 0.073\) (7.3% as a decimal)
- \(n = 18\)
Now plug these values into the formula:
\[
A = 5000(1 + 0.073)^{18}
\]
\[
A = 5000(1.073)^{18}
\]
First, we calculate \( (1.073)^{18} \):
\[
(1.073)^{18} \approx 3.58315
\]
Now multiply by the principal amount:
\[
A \approx 5000 \times 3.58315 \approx 17915.75
\]
Rounding to two decimal places, the total value of the college fund after 18 years is approximately **$17,915.75**.
Next, to find the interest earned on the account, we subtract the principal from the total amount:
\[
\text{Interest} = A - P
\]
\[
\text{Interest} = 17915.75 - 5000 = 12915.75
\]
So, the interest earned on the account is approximately **$12,915.75**.
Final answers:
- The final amount of the college fund is **$17,915.75**.
- The amount of interest earned on the account is **$12,915.75**.
\[
A = P(1 + r)^n
\]
where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of years the money is invested or borrowed for.
In this case:
- \(P = 5000\)
- \(r = 0.073\) (7.3% as a decimal)
- \(n = 18\)
Now plug these values into the formula:
\[
A = 5000(1 + 0.073)^{18}
\]
\[
A = 5000(1.073)^{18}
\]
First, we calculate \( (1.073)^{18} \):
\[
(1.073)^{18} \approx 3.58315
\]
Now multiply by the principal amount:
\[
A \approx 5000 \times 3.58315 \approx 17915.75
\]
Rounding to two decimal places, the total value of the college fund after 18 years is approximately **$17,915.75**.
Next, to find the interest earned on the account, we subtract the principal from the total amount:
\[
\text{Interest} = A - P
\]
\[
\text{Interest} = 17915.75 - 5000 = 12915.75
\]
So, the interest earned on the account is approximately **$12,915.75**.
Final answers:
- The final amount of the college fund is **$17,915.75**.
- The amount of interest earned on the account is **$12,915.75**.