Question

How much money will there be in an account at the end of 9 years if ​$17000 is deposited at 3​% interest compounded semiannually question mark ​(Assume no withdrawals are​ made.)
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Question content area right
Part 1
The amount after 9 years will be ​$
  
enter your response here.
​(Round to the nearest cent as​ needed.)

Answer 1

To calculate the future value of an investment with compound interest, we can use the formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

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 times that interest is compounded per year.
- \(t\) = the number of years the money is invested for.

Given:
- \(P = 17000\)
- \(r = 0.03\) (3% as a decimal)
- \(n = 2\) (because interest is compounded semiannually)
- \(t = 9\)

Now plug the values into the formula:

\[
A = 17000 \left(1 + \frac{0.03}{2}\right)^{2 \cdot 9}
\]

Calculating step by step:

1. Calculate \(\frac{r}{n}\):

\[
\frac{0.03}{2} = 0.015
\]

2. Calculate \(nt\):

\[
2 \cdot 9 = 18
\]

3. Substitute these values back into the formula:

\[
A = 17000 \left(1 + 0.015\right)^{18}
\]

4. Calculate \(1 + 0.015\):

\[
1 + 0.015 = 1.015
\]

5. Now calculate \(A\):

\[
A = 17000 \cdot (1.015)^{18}
\]

First, calculate \((1.015)^{18}\):

\[
(1.015)^{18} \approx 1.346855007
\]

6. Now, multiply by the principal:

\[
A \approx 17000 \cdot 1.346855007 \approx 22995.53512
\]

7. Round to the nearest cent:

\[
A \approx 22995.54
\]

Thus, the amount after 9 years will be approximately **$22995.54**.