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 = $43,000, r = 6
3
4
%, t = 5,
 compounded quarterly

The formula for the accumulated amount $ A $ when interest is compounded periodically is given by:

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

Where:
- $ P $ is the principal,
- $ r $ is the annual interest rate (as a decimal),
- $ t $ is the time in years,
- $ n $ is the number of times the interest is compounded per year.

In this case:
- $ P = 43,000 $,
- $ r = 6 \frac{3}{4} \% = 0.0675 $ (as a decimal),
- $ t = 5 $ years,
- Compounded quarterly means $ n = 4 $.

### Step-by-step calculation:

1. Convert the interest rate to a decimal:  
   $ r = 6 \frac{3}{4} \% = 0.0675 $.

2. Plug the values into the formula:
   \[
   A = 43,000 \left(1 + \frac{0.0675}{4}\right)^{4 \times 5}
   \]
   Simplify further:
   \[
   A = 43,000 \left(1 + 0.016875\right)^{20}
   \]
   \[
   A = 43,000 \left(1.016875\right)^{20}
   \]

3. Calculate $ \left(1.016875\right)^{20} $.

4. Multiply the result by 43,000 to get the final accumulated amount.

Let me calculate that.The accumulated amount $ A $ after 5 years with a principal of $43,000, an interest rate of 6.75% compounded quarterly, is approximately **$60,092.44**.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What would happen if the interest was compounded monthly instead of quarterly?
2. How does the accumulated amount change if the interest rate increases by 1%?
3. How does simple interest compare to compound interest in this scenario?
4. What is the formula for continuously compounded interest?
5. How would the accumulated amount change if the investment period was 10 years?

**Tip**: Compound interest grows faster than simple interest because it adds interest on top of the previously earned interest!

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 = $43,000, r = 6 3/4 %, t = 5, compounded quarterly.

Find the accumulated amount A for an investment of $43,000 at an annual interest rate of 6.75% compounded quarterly over 5 years. Use the compound interest formula to calculate exponential growth. Suitable for Grades 9-12.

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