https://file.aiquickdraw.com/mathbot/file/1729266092023zwu4ursf.jpg

The formula given in the image is for compound interest:

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

Where:
- $ A(t) $ is the amount of money accumulated after $ t $ years, including interest.
- $ P $ is the principal amount (initial investment).
- $ r $ is the annual interest rate (as a decimal).
- $ n $ is the number of times the interest is compounded per year.
- $ t $ is the time in years.

In this case, we are given:
- $ P = 2000 $ (the principal is $2,000),
- $ r = 0.07 $ (7% interest rate),
- $ n = 2 $ (compounded twice a year),
- $ t = 10 $ years.

We can plug these values into the formula:

\[
A(10) = 2000 \left( 1 + \frac{0.07}{2} \right)^{2 \times 10}
\]

Now let's calculate this step by step.

1. Calculate $ \frac{0.07}{2} = 0.035 $.
2. Add 1 to it: $ 1 + 0.035 = 1.035 $.
3. Now raise this to the power of $ 2 \times 10 = 20 $: $ 1.035^{20} \approx 2.030377 $.
4. Multiply by the principal $ 2000 $: 

\[
A(10) \approx 2000 \times 2.030377 = 4060.75
\]

Thus, the amount accumulated after 10 years is approximately **$4,060.75**.

Would you like further details on any part of the solution? Here are some related questions:

1. How does the frequency of compounding affect the final amount?
2. What would the accumulated amount be with a different interest rate?
3. How would changing the investment duration impact the result?
4. What is the difference between compound interest and simple interest?
5. How can this formula be adjusted for continuous compounding?

**Tip**: The more frequent the compounding, the greater the accumulated amount, even if the interest rate and time period remain constant.

Determine the amount of money accumulated after 10 years if $2,000 is invested in an account that pays 7% interest compounded twice a year. Round to the nearest cent.

Calculate the accumulated amount after 10 years for an initial investment of $2,000 at 7% interest, compounded semiannually. Learn how to apply the compound interest formula step-by-step and round the result to the nearest cent. Suitable for high school students learning about exponential growth.

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