Suppose the present value of a continuous income stream over the next 9 years is given by
P = 4,000
9

0
te−0.05t dt,
with P in dollars and t in years. Find the present value (in dollars). (Round your answer to the nearest cent.)
$ 

The given equation for the present value $ P $ of a continuous income stream is:

\[
P = 4000 \int_0^9 t e^{-0.05t} \, dt
\]

This is a definite integral that we need to solve in order to find the present value.

We'll break this down into steps:

1. **Integration by Parts**: Since we have the product of two functions $ t $ and $ e^{-0.05t} $, we'll use integration by parts. Let's use the formula:
   
\[
\int u \, dv = uv - \int v \, du
\]

We'll set:
- $ u = t $, so $ du = dt $
- $ dv = e^{-0.05t} \, dt $, so $ v = \frac{-1}{0.05} e^{-0.05t} = -20 e^{-0.05t} $

Now, let's apply this and compute the definite integral. I'll do this step for you.

The present value $ P $ of the continuous income stream is approximately:

\[
P \approx 120,702.69
\]

Would you like further details on the integration process or have any questions?

Here are some related questions you might find interesting:
1. What is the general formula for calculating the present value of a continuous income stream?
2. How does the discount rate affect the present value in continuous compounding?
3. What role does integration by parts play in solving these types of problems?
4. Could we compute present value for different time horizons, e.g., over 5 or 10 years?
5. How would this problem change if the exponent in the decay function were different?

**Tip**: When solving integrals involving exponential decay and polynomial functions, integration by parts is often the most efficient approach.

Suppose the present value of a continuous income stream over the next 9 years is given by P = 4000 ∫ 0 to 9 (t * e^(-0.05t)) dt. Find the present value (in dollars). (Round your answer to the nearest cent.)

Find the present value of a continuous income stream using integration by parts. This problem involves solving a definite integral with exponential decay, covering key calculus concepts like integration by parts. Ideal for college-level calculus students.

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