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Part 1
The future value at 4.75​% ​interest, compounded continuously for 6 ​years, of the continuous income stream with rate of flow ​f(t)equals2 comma 025 e Superscript negative 0.02 t​, is ​$13 comma 285. Compute the interest earned.
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Part 1
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enter your response here ​(Type an integer or a​ decimal.)

To find the interest earned, we first need to determine the initial value of the income stream and then subtract it from the future value.

The future value $ FV $ of a continuous income stream with rate of flow $ f(t) $, compounded continuously at a rate $ r $ over a time period $ T $, is given by:

\[
FV = e^{rT} \int_0^T f(t) \, dt
\]

Given:

- $ r = 4.75\% = 0.0475 $
- $ T = 6 $ years
- $ f(t) = 2025e^{-0.02t} $
- Future value $ FV = 13,285 $

### Step 1: Compute the initial value $ IV $
The initial value is the integral of the rate of flow over the time period $ T $:

\[
IV = \int_0^6 2025e^{-0.02t} \, dt
\]

Let's compute this integral:

\[
IV = 2025 \int_0^6 e^{-0.02t} \, dt
\]

The integral of $ e^{-0.02t} $ is:

\[
\int e^{-0.02t} \, dt = \frac{-1}{0.02} e^{-0.02t} = -50e^{-0.02t}
\]

So:

\[
IV = 2025 \left[-50e^{-0.02t}\right]_0^6
\]

Evaluate this:

\[
IV = 2025 \left[-50(e^{-0.12} - 1)\right]
\]

Simplify:

\[
IV = 2025 \times 50 \times (1 - e^{-0.12})
\]

\[
IV = 101,250 \times (1 - e^{-0.12})
\]

Calculate $ e^{-0.12} $ and subtract it from 1, then multiply by 101,250.

### Step 2: Calculate the interest earned
The interest earned is the difference between the future value and the initial value:

\[
\text{Interest Earned} = FV - IV
\]

Let's compute these values.The interest earned is approximately **$1,835.69**.

Would you like any further details or have additional questions?

Here are 8 related questions:
1. How does continuous compounding differ from other types of compounding?
2. What is the significance of the rate of flow $ f(t) $ in the context of continuous income streams?
3. How do you calculate the future value for different compounding intervals?
4. What would the interest earned be if the compounding rate was 5% instead of 4.75%?
5. How does changing the time period $ T $ affect the future value and interest earned?
6. Can the rate of flow $ f(t) $ be a non-exponential function? How would that change the calculations?
7. What if the income stream is not continuous but periodic? How would the future value be computed?
8. How is the formula for continuous compounding derived?

**Tip:** Continuous compounding leads to the highest possible future value for a given interest rate, as interest is added at every possible instant.

Learn how to calculate interest earned on a continuous income stream compounded at 4.75% interest for 6 years. Includes step-by-step solution with details on exponential functions and integral calculus. Suitable for advanced high school students and learners interested in financial mathematics.

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