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The image asks to investigate continuous growth using the formula for continuous compound interest: 

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

Where:
- $ A(t) $ is the amount after time $ t $,
- $ P $ is the initial principal (initial deposit),
- $ r $ is the interest rate (expressed as a decimal),
- $ t $ is the time in years, and
- $ e $ is Euler’s number (approximately 2.71828).

Let's break down the problem:

### Given Information:
- Initial deposit $ P = 11,500 $ dollars.
- Annual interest rate $ r = 7.1\% = 0.071 $ (as a decimal).
- Time $ t = 25 $ years.

#### Part (a): Compounded Continuously
For continuous compounding, we use the formula:

\[
A(t) = 11,500 \cdot e^{(0.071 \cdot 25)}
\]

Let me calculate this for you.

\[
A(25) = 11,500 \cdot e^{(0.071 \cdot 25)} \approx 11,500 \cdot e^{1.775} \approx 11,500 \cdot 5.9002 \approx 67,852.30
\]

So, after 25 years, the account will be worth approximately **$67,852.30** if compounded continuously.

#### Part (b): Compounded Monthly
For monthly compounding, we use the standard compound interest formula:

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

Where:
- $ n = 12 $ (since it's compounded monthly).

The formula becomes:

\[
A(25) = 11,500 \left(1 + \frac{0.071}{12}\right)^{12 \cdot 25}
\]

Let's calculate it:

\[
A(25) = 11,500 \left(1 + 0.005917\right)^{300} \approx 11,500 \cdot 5.8461 \approx 67,230.15
\]

So, after 25 years, the account will be worth approximately **$67,230.15** if compounded monthly.

### Summary:
- **Compounded Continuously**: $67,852.30
- **Compounded Monthly**: $67,230.15

Would you like more details on any of these calculations, or have further questions?

### Related Questions:
1. How would the value change if the interest rate was 5% instead of 7.1%?
2. What would the account balance be if the time period was 30 years instead of 25 years?
3. How does the frequency of compounding (e.g., weekly or daily) affect the final amount?
4. Can you derive the continuous compound interest formula from basic principles of limits?
5. How does continuous compounding compare to quarterly or yearly compounding?

### Tip:
Continuous compounding often yields the highest amount of interest due to the constant application of the growth rate, but the difference diminishes as the compounding frequency increases.

Suppose an investment account is opened with an initial deposit of $11,500.00 earning 7.1% interest compounded continuously. How much will the account be worth after 25 years? How much will the account be worth after 25 years if it were compounded monthly instead?

Learn how to calculate the future value of an investment with $11,500 initial deposit, 7.1% annual interest rate, compounded continuously or monthly over 25 years. This example demonstrates continuous compound interest and monthly compounding with key formulas.

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