The doubling time of an investment with continuous compound interest is 12.4 years.  If the investment is worth ​$25 comma 000 ​today, how much will it be worth 5 years from​ now?


To find the value of an investment under continuous compound interest after 5 years, we'll use the continuous compounding formula:

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

Where:
- $ A $ is the amount of the investment after time $ t $.
- $ P $ is the principal (initial investment), which is $25,000.
- $ r $ is the interest rate (which we will calculate).
- $ t $ is the time in years (in this case, 5 years).
- $ e $ is Euler's number (approximately 2.71828).

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

The doubling time formula for continuous compounding is:

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

Given that the doubling time is 12.4 years, we can solve for $ r $:

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

Solving for $ r $:

\[
r = \frac{\ln(2)}{12.4}
\]

### Step 2: Calculate $ A $ after 5 years

Once we have the interest rate $ r $, we can substitute the values into the compound interest formula to find how much the investment will be worth after 5 years. Let me calculate this.The investment, which is worth $25,000 today, will be worth approximately **$33,061.54** after 5 years under continuous compounding.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does continuous compounding differ from other compounding methods?
2. What is the doubling time for an investment with different interest rates?
3. How does the time period affect the future value in continuous compounding?
4. How would this calculation change if the interest were compounded annually instead of continuously?
5. What effect would changing the initial investment amount have on the future value?

**Tip:** The formula for continuous compounding can be used for any time period, so it's especially useful for long-term investments.

The doubling time of an investment with continuous compound interest is 12.4 years. If the investment is worth $25,000 today, how much will it be worth 5 years from now?

Learn how to calculate the future value of a $25,000 investment with continuous compound interest, doubling in 12.4 years. Understand the use of the formula A = P * e^(rt) to determine that the investment will grow to approximately $33,061.54 in 5 years. This is a practical example of exponential growth, suitable for students in Grades 10-12.

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