What rate of interest compounded semiannually doubles an investment in 8 years?  Round your answer to 2 decimal places.

To determine the rate of interest that doubles an investment in 8 years with semiannual compounding, we can use the compound interest formula:

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

Where:
- \( A \) is the final amount (which is double the initial investment, so \( A = 2P \)),
- \( P \) is the principal (initial investment),
- \( r \) is the annual interest rate (which we need to find),
- \( n \) is the number of times the interest is compounded per year (semiannually, so \( n = 2 \)),
- \( t \) is the time in years (8 years).

Given:
\[
A = 2P, \quad n = 2, \quad t = 8
\]

We can plug these values into the compound interest formula and solve for \( r \):

\[
2P = P \left(1 + \frac{r}{2}\right)^{2 \cdot 8}
\]

Simplifying the equation:

\[
2 = \left(1 + \frac{r}{2}\right)^{16}
\]

Now, we solve for \( r \). First, take the 16th root of both sides:

\[
\left(1 + \frac{r}{2}\right) = 2^{\frac{1}{16}}
\]

Let me calculate this.We have:

\[
1 + \frac{r}{2} = 1.04427
\]

Now, subtract 1 from both sides:

\[
\frac{r}{2} = 0.04427
\]

Multiply by 2 to solve for \( r \):

\[
r = 0.08854
\]

Therefore, the annual interest rate is approximately \( r = 8.85\% \).

Would you like further details on the calculation or explanation?

Here are 5 follow-up questions to explore this further:
1. How would the interest rate change if the compounding period was quarterly instead of semiannually?
2. Can you derive a formula for the interest rate if the investment triples in 8 years?
3. How does continuous compounding affect the interest rate for the same scenario?
4. What would the interest rate be if the investment doubled in 10 years with semiannual compounding?
5. How does the time period affect the interest rate needed for doubling an investment?

**Tip:** When solving compound interest problems, always match the number of compounding periods with the interest rate and time frame for accurate results.

What rate of interest compounded semiannually doubles an investment in 8 years? Round your answer to 2 decimal places.

Discover how to calculate the semiannual interest rate needed to double an investment in 8 years. This problem utilizes the compound interest formula and covers key concepts such as exponential growth and algebraic manipulation. Suitable for high school students in Grades 10-12.

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