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 Martin's college fund after 18 years with an initial deposit (principal) 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:
- \(A\) = the amount of money accumulated after n years, including interest.
- \(P\) = the principal amount (the initial amount of money).
- \(r\) = annual interest rate (decimal).
- \(n\) = number of years the money is invested or borrowed.
### Given Values:
- \(P = 5000\)
- \(r = 0.073\) (7.3% as a decimal)
- \(n = 18\)
### Step 1: Calculate the total value after 18 years
Plugging these values into the formula:
\[
A = 5000(1 + 0.073)^{18}
\]
\[
A = 5000(1.073)^{18}
\]
Now, calculate \( (1.073)^{18} \):
\[
(1.073)^{18} \approx 3.588
\]
Now, multiply by the principal:
\[
A \approx 5000 \times 3.588 \approx 17940
\]
### Final Amount
Thus, the total value of the account after 18 years is approximately:
\[
A \approx 17940.20
\]
### Step 2: Calculate the interest earned
To find the interest earned, subtract the principal from the total amount:
\[
\text{Interest} = A - P = 17940.20 - 5000
\]
\[
\text{Interest} \approx 12940.20
\]
### Summary
- The final amount of the college fund is **$17,940.20**.
- The amount of interest earned on the account is **$12,940.20**.
### Final Answers:
- Final Amount: $17,940.20
- Interest Earned: $12,940.20
\[
A = P(1 + r)^n
\]
Where:
- \(A\) = the amount of money accumulated after n years, including interest.
- \(P\) = the principal amount (the initial amount of money).
- \(r\) = annual interest rate (decimal).
- \(n\) = number of years the money is invested or borrowed.
### Given Values:
- \(P = 5000\)
- \(r = 0.073\) (7.3% as a decimal)
- \(n = 18\)
### Step 1: Calculate the total value after 18 years
Plugging these values into the formula:
\[
A = 5000(1 + 0.073)^{18}
\]
\[
A = 5000(1.073)^{18}
\]
Now, calculate \( (1.073)^{18} \):
\[
(1.073)^{18} \approx 3.588
\]
Now, multiply by the principal:
\[
A \approx 5000 \times 3.588 \approx 17940
\]
### Final Amount
Thus, the total value of the account after 18 years is approximately:
\[
A \approx 17940.20
\]
### Step 2: Calculate the interest earned
To find the interest earned, subtract the principal from the total amount:
\[
\text{Interest} = A - P = 17940.20 - 5000
\]
\[
\text{Interest} \approx 12940.20
\]
### Summary
- The final amount of the college fund is **$17,940.20**.
- The amount of interest earned on the account is **$12,940.20**.
### Final Answers:
- Final Amount: $17,940.20
- Interest Earned: $12,940.20