Kavi needs 530.200 tor a future protect. He can investosoo now at an annual rate or 11%, compounded semiannually. Assuming that no withdrawals are made, how long will it take for him to have enough money for his project?
Do not round anv intermediate combutations, and round vour answer to the nearest hundredth

To determine how long it will take for Kavi to have enough money for his project, we can use the formula for compound interest:

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

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

In this problem:
- $ A = 530,200 $
- $ P = 200,000 $
- $ r = 0.11 $
- $ n = 2 $ (since the interest is compounded semiannually)

We need to solve for $ t $.

First, let's substitute the known values into the compound interest formula:

\[ 530,200 = 200,000 \left(1 + \frac{0.11}{2}\right)^{2t} \]

Simplify inside the parentheses:

\[ 530,200 = 200,000 \left(1 + 0.055\right)^{2t} \]
\[ 530,200 = 200,000 \left(1.055\right)^{2t} \]

Divide both sides by 200,000 to isolate the exponential term:

\[ \frac{530,200}{200,000} = \left(1.055\right)^{2t} \]
\[ 2.651 = \left(1.055\right)^{2t} \]

To solve for $ t $, take the natural logarithm of both sides:

\[ \ln(2.651) = \ln\left( (1.055)^{2t} \right) \]

Using the logarithm power rule $\ln(a^b) = b \cdot \ln(a)$:

\[ \ln(2.651) = 2t \cdot \ln(1.055) \]

Solve for $ t $:

\[ t = \frac{\ln(2.651)}{2 \cdot \ln(1.055)} \]

Calculate the values:

\[ \ln(2.651) \approx 0.974 \]
\[ \ln(1.055) \approx 0.0536 \]

\[ t = \frac{0.974}{2 \cdot 0.0536} \]
\[ t = \frac{0.974}{0.1072} \]
\[ t \approx 9.09 \]

Therefore, it will take Kavi approximately 9.09 years to have enough money for his project.

Would you like more details or have any questions about this solution?

Here are some other questions you might consider asking:
1. How does changing the interest rate affect the time required?
2. What if Kavi could make additional contributions periodically?
3. How is compound interest different from simple interest?
4. Can you show the steps for converting the logarithm to a decimal value?
5. What is the formula for continuous compounding interest?

**Tip:** Always double-check your calculations, especially when dealing with exponentiation and logarithms, as small errors can significantly affect the final result.

Learn how to calculate the time required for an investment to reach a future value using the compound interest formula. This step-by-step solution demonstrates how to determine the time needed for Kavi's investment of $200,000 to grow to $530,200 at an annual interest rate of 11% compounded semiannually.

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