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 $
.
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 find the total value of Martin's college fund after 18 years with an initial 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:
- \(A\) 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.
Given:
- \(P = 5000\)
- \(r = 0.073\) (7.3% expressed as a decimal)
- \(n = 18\)
Now, we can substitute these values into the formula:
\[
A = 5000(1 + 0.073)^{18}
\]
Calculating \(1 + 0.073\):
\[
1 + 0.073 = 1.073
\]
Now raising this to the power of 18:
\[
A = 5000 \times (1.073)^{18}
\]
Calculating \((1.073)^{18}\):
\[
(1.073)^{18} \approx 3.634
\]
Now, substituting this value back into the equation to find \(A\):
\[
A \approx 5000 \times 3.634 \approx 18170
\]
So, the total amount in the college fund after 18 years is approximately:
\[
A \approx 18170.00
\]
Next, to find the amount of interest earned:
\[
\text{Interest} = A - P
\]
Substituting the values we have:
\[
\text{Interest} \approx 18170.00 - 5000 = 13170.00
\]
Thus, the final amounts are:
- The final amount of the college fund is approximately **$18,170.00**.
- The amount of interest earned on the account is approximately **$13,170.00**.
Final answers:
The final amount of the college fund is **$18170.00**, and the amount of interest earned on the account is **$13170.00**.
\[
A = P(1 + r)^n
\]
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.
Given:
- \(P = 5000\)
- \(r = 0.073\) (7.3% expressed as a decimal)
- \(n = 18\)
Now, we can substitute these values into the formula:
\[
A = 5000(1 + 0.073)^{18}
\]
Calculating \(1 + 0.073\):
\[
1 + 0.073 = 1.073
\]
Now raising this to the power of 18:
\[
A = 5000 \times (1.073)^{18}
\]
Calculating \((1.073)^{18}\):
\[
(1.073)^{18} \approx 3.634
\]
Now, substituting this value back into the equation to find \(A\):
\[
A \approx 5000 \times 3.634 \approx 18170
\]
So, the total amount in the college fund after 18 years is approximately:
\[
A \approx 18170.00
\]
Next, to find the amount of interest earned:
\[
\text{Interest} = A - P
\]
Substituting the values we have:
\[
\text{Interest} \approx 18170.00 - 5000 = 13170.00
\]
Thus, the final amounts are:
- The final amount of the college fund is approximately **$18,170.00**.
- The amount of interest earned on the account is approximately **$13,170.00**.
Final answers:
The final amount of the college fund is **$18170.00**, and the amount of interest earned on the account is **$13170.00**.