The doubling time of an investment with continuous compound interest is 12.2years.  If the investment is worth ​$20000 ​today, how much will it be worth 7 years from​ now? Round to the nearest cent

To solve this problem, we can use the formula for continuous compound interest:

\[
A = P e^{rt}
\]

Where:
- $ A $ is the amount of money in the account after time $ t $.
- $ P $ is the initial investment.
- $ r $ is the continuous interest rate.
- $ t $ is the time in years.
- $ e $ is the base of the natural logarithm (approximately 2.71828).

### Step 1: Find the interest rate $ r $
We know the doubling time is 12.2 years. The formula for doubling time with continuous compounding is:

\[
t_{\text{double}} = \frac{\ln(2)}{r}
\]

Given that $ t_{\text{double}} = 12.2 $, we can solve for $ r $:

\[
12.2 = \frac{\ln(2)}{r}
\]

\[
r = \frac{\ln(2)}{12.2}
\]

Now, I'll calculate $ r $.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can still guide you through solving the problem manually:

### Step 1: Find the interest rate $ r $

We already set up the equation:

\[
r = \frac{\ln(2)}{12.2}
\]

- $ \ln(2) \approx 0.6931 $
  
So,

\[
r = \frac{0.6931}{12.2} \approx 0.0568
\]

### Step 2: Calculate the future value of the investment

Now, we use the continuous compounding formula to find the amount after 7 years:

\[
A = 20000 \times e^{(0.0568 \times 7)}
\]

Calculating the exponent and multiplying:

\[
A \approx 20000 \times e^{0.3976} \approx 20000 \times 1.4884 \approx 29,767.87
\]

Thus, the investment will be worth approximately **$29,767.87** after 7 years.

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Do you need more details on any part? Feel free to ask!

Here are 5 related questions for practice:
1. What is the formula for compound interest (non-continuous)?
2. How does the doubling time change if the interest rate doubles?
3. How would the calculation change if the time was 10 years instead of 7?
4. What would the future value be for an investment of $50,000 with the same interest rate?
5. What is the difference between continuous and periodic compounding?

**Tip:** Continuous compounding grows an investment at the fastest possible rate, compared to periodic compounding.

The doubling time of an investment with continuous compound interest is 12.2 years. If the investment is worth $20000 today, how much will it be worth 7 years from now? Round to the nearest cent.

Learn how to calculate the future value of a $20,000 investment over 7 years with continuous compounding and a doubling time of 12.2 years. This detailed explanation covers exponential growth and logarithmic calculations, suitable for high school students.

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