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 = $250,000, r = 5%, t = 6, compounded daily
A = $

To find the accumulated amount $ A $ when the principal $ P $ is invested at an interest rate $ r $ for $ t $ years, compounded daily, we use the compound interest formula:

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

Where:
- $ A $ is the accumulated amount,
- $ P = 250,000 $ is the principal,
- $ r = 0.05 $ is the annual interest rate (5%),
- $ t = 6 $ is the number of years,
- $ n = 365 $ is the number of times the interest is compounded per year (daily compounding).

Let's calculate it step by step:

1. Substitute the given values into the formula:

\[
A = 250,000 \left( 1 + \frac{0.05}{365} \right)^{365 \times 6}
\]

2. Simplify the fraction inside the parentheses:

\[
\frac{0.05}{365} = 0.000136986
\]

3. Now, compute the power term $ 365 \times 6 = 2190 $.

4. Add 1 to the fraction:

\[
A = 250,000 \left( 1 + 0.000136986 \right)^{2190}
= 250,000 \left( 1.000136986 \right)^{2190}
\]

5. Calculate the exponentiation:

\[
1.000136986^{2190} \approx 1.349858807
\]

6. Multiply by the principal:

\[
A = 250,000 \times 1.349858807 \approx 337,464.70
\]

Thus, the accumulated amount $ A $ is approximately **$337,464.70**.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the accumulated amount if $ t = 10 $ years?
2. How would the accumulated amount change with quarterly compounding?
3. What happens to the accumulated amount if the interest rate increases to 6%?
4. How does daily compounding compare to monthly compounding for the same $ t $ and $ r $?
5. What is the formula for continuously compounded interest?

**Tip:** Compounding frequency has a significant effect on the accumulated interest, with more frequent compounding leading to higher amounts.

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 = $250,000, r = 5%, t = 6, compounded daily A = $

Learn how to calculate the accumulated amount for an investment of $250,000 at an annual interest rate of 5%, compounded daily over 6 years. This detailed solution covers compound interest formulas and financial mathematics. Suitable for students in Grades 10-12.

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