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.
(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.
(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 investment of $5,000 and an annual compound interest rate of 7.3%, we can use the formula for compound interest:
\[
A = P(1 + r)^t
\]
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).
- \(t\) is the number of years the money is invested or borrowed.
Given:
- \(P = 5000\)
- \(r = 0.073\) (7.3% expressed as a decimal)
- \(t = 18\)
Plugging in the values:
\[
A = 5000(1 + 0.073)^{18}
\]
Calculating \(1 + 0.073\):
\[
1 + 0.073 = 1.073
\]
Now raising this to the power of 18:
\[
1.073^{18} \approx 3.617
\]
Now, calculate \(A\):
\[
A = 5000 \times 3.617 \approx 18085
\]
Now rounding to two decimal places, the total value of the college fund after 18 years is:
\[
A \approx 18085.00
\]
Next, we need to calculate the amount of interest earned on the account. The interest earned is the final amount minus the principal amount:
\[
\text{Interest} = A - P
\]
\[
\text{Interest} = 18085.00 - 5000 = 13085.00
\]
Now rounding to two decimal places, the amount of interest earned is:
\[
\text{Interest} \approx 13085.00
\]
Therefore, the final amounts are:
- The final amount of the college fund is **$18,085.00**.
- The amount of interest earned on the account is **$13,085.00**.
\[
A = P(1 + r)^t
\]
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).
- \(t\) is the number of years the money is invested or borrowed.
Given:
- \(P = 5000\)
- \(r = 0.073\) (7.3% expressed as a decimal)
- \(t = 18\)
Plugging in the values:
\[
A = 5000(1 + 0.073)^{18}
\]
Calculating \(1 + 0.073\):
\[
1 + 0.073 = 1.073
\]
Now raising this to the power of 18:
\[
1.073^{18} \approx 3.617
\]
Now, calculate \(A\):
\[
A = 5000 \times 3.617 \approx 18085
\]
Now rounding to two decimal places, the total value of the college fund after 18 years is:
\[
A \approx 18085.00
\]
Next, we need to calculate the amount of interest earned on the account. The interest earned is the final amount minus the principal amount:
\[
\text{Interest} = A - P
\]
\[
\text{Interest} = 18085.00 - 5000 = 13085.00
\]
Now rounding to two decimal places, the amount of interest earned is:
\[
\text{Interest} \approx 13085.00
\]
Therefore, the final amounts are:
- The final amount of the college fund is **$18,085.00**.
- The amount of interest earned on the account is **$13,085.00**.