suppose $15000 is invested at 7% interest compounded continuously. How long will it take for the investment to double? Round your answer to two decimal places.

To determine how long it will take for the investment to double when interest is compounded continuously, we can use the continuous compound interest formula:

\[
A(t) = P \cdot e^{rt}
\]

Where:
- $ A(t) $ is the amount after time $ t $,
- $ P $ is the initial principal (in this case, $15,000),
- $ r $ is the interest rate (7% or 0.07),
- $ t $ is the time in years.

We need to find the time $ t $ when the investment doubles, meaning $ A(t) = 2P $:

\[
2P = P \cdot e^{rt}
\]

Dividing both sides by $ P $:

\[
2 = e^{0.07t}
\]

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

\[
\ln(2) = \ln(e^{0.07t})
\]

Using the property of logarithms $ \ln(e^x) = x $, we have:

\[
\ln(2) = 0.07t
\]

Now, solve for $ t $:

\[
t = \frac{\ln(2)}{0.07}
\]

Calculate $ t $:

\[
t \approx \frac{0.693147}{0.07} \approx 9.9021
\]

Rounded to two decimal places, the time it will take for the investment to double is approximately:

\[
\boxed{9.90 \text{ years}}
\]

Would you like further details or have any questions?

### Related Questions
1. What would be the time required to double the investment at an interest rate of 5%?
2. How would the time change if the interest were compounded annually instead of continuously?
3. How much will the investment be worth after 15 years?
4. How does the doubling time change with different interest rates?
5. If the initial investment were $10,000, how would that affect the doubling time?
6. How can the rule of 72 be used to estimate the doubling time?
7. What is the effect of increasing the interest rate by 1% on the doubling time?
8. How long will it take for the investment to triple at the same interest rate?

### Tip
When solving exponential equations, logarithms are often the key tool for isolating the variable of interest.

Learn how to calculate the time required for an investment to double with continuous compound interest. Use the continuous compound interest formula to solve the problem where $15,000 is invested at 7% interest compounded continuously. Get a detailed solution and rounded answer to two decimal places.

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