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The problem asks for the three-year future value of a company's income stream, where profits of $40,000 per year are continuously invested at a 5.4% annual percentage rate (APR) compounded continuously.

The future value of continuously compounded investments is calculated using the formula:

\[
FV = P \times e^{rt}
\]

Where:
- $ P $ is the amount invested,
- $ r $ is the annual interest rate (in decimal form),
- $ t $ is the time in years,
- $ e $ is Euler's number (approximately 2.71828).

However, since this is an income stream where $40,000 is earned each year over 3 years, we use a **future value of continuous annuity formula**:

\[
FV = \int_0^T R \times e^{r(T - t)} dt
\]

Where:
- $ R $ is the annual profit ($40,000),
- $ r $ is the interest rate (5.4% = 0.054),
- $ T $ is the total duration (3 years).

Now, solving for this specific case:

\[
FV = 40000 \times \left(\frac{e^{0.054 \times 3} - 1}{0.054}\right)
\]

Let me calculate the future value for you.The future value of the company's income stream, with continuous compounding at a 5.4% APR for 3 years, is approximately **$130,266.85**.

This matches the second option in the choices provided.

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would the future value change if the interest rate were higher, say 6%?
2. What if the investment period was extended to 5 years?
3. How would the result differ if the profits were not invested continuously but annually?
4. What happens to the future value if the company starts with an initial lump sum investment?
5. How would inflation affect the real value of this income stream over time?

**Tip:** For continuous compounding, small changes in the interest rate or investment duration can have a large impact on the future value, due to the exponential growth factor.

A company expects profits of $40,000 per year for 3 years. If the profits will be continuously invested in an account bearing 5.4% APR compounded continuously, what is the three-year future value of this income stream?

Calculate the three-year future value of a $40,000 annual income stream with continuous compounding at 5.4% APR. Learn about continuous compounding and the future value of annuities using key formulas and exponential growth. Suitable for financial mathematics students.

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