Find the accumulated amount A if the principal P is invested at the interest rate of r/year for t years. (Use a 365-day year. Round your answer to the nearest cent.)
P = $40,000, r = 8
3
4
%, t = 4,
 compounded quarterly
A = $

We can use the compound interest formula to find the accumulated amount $ A $:

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

Where:
- $ A $ is the accumulated amount (the future value),
- $ P $ is the principal amount (initial investment),
- $ r $ is the annual interest rate (in decimal form),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the time the money is invested for (in years).

### Given values:
- $ P = 40,000 $
- $ r = 8 \frac{3}{4} \% = 8.75\% = 0.0875 $ (in decimal form),
- $ n = 4 $ (since it's compounded quarterly),
- $ t = 4 $ years.

Now, substitute these values into the formula:

\[
A = 40,000 \left(1 + \frac{0.0875}{4}\right)^{4 \times 4}
\]

\[
A = 40,000 \left(1 + 0.021875\right)^{16}
\]

\[
A = 40,000 \left(1.021875\right)^{16}
\]

Now, let's calculate the result:

\[
A = 40,000 \times 1.421613
\]

\[
A \approx 56,864.52
\]

### Final Answer:
The accumulated amount $ A $ is approximately **$56,864.52**.

---

Would you like further details or clarifications?

Here are five related questions to consider:
1. What is the difference between simple and compound interest?
2. How does the frequency of compounding affect the accumulated amount?
3. What happens if the interest rate changes halfway through the investment period?
4. How do you calculate continuous compounding interest?
5. Can you derive the formula for compound interest?

**Tip:** Increasing the frequency of compounding (e.g., from yearly to quarterly) will generally result in a higher accumulated amount.

Find the accumulated amount A if the principal P is invested at the interest rate of r/year for t years. (Use a 365-day year. Round your answer to the nearest cent.) P = $40,000, r = 8 3/4%, t = 4, compounded quarterly. A = $

Learn how to calculate the accumulated amount for a principal of $40,000 invested at an 8.75% interest rate, compounded quarterly for 4 years. This detailed guide covers the compound interest formula and provides a step-by-step solution. Suitable for high school students.

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