Owen sells his business and receives yearly payments on a sliding scale according to the function P(t) = 70000-10000t. if this income is treated as a continuous stream and deposited into an account that earns an annual interest rate of 3.25% compounded continuously, How much will the account be worth after 7 years?

The problem requires calculating the future value of a continuous stream of income deposited into an account with interest compounded continuously.

Here, the yearly payments $ P(t) $ decrease over time according to the function $ P(t) = 70,000 - 10,000t $, and the income is deposited into an account that earns 3.25% interest, compounded continuously. The total value of the account after 7 years is calculated using the formula for the future value of a continuously compounded income stream:

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

Where:
- $ P(t) $ is the payment function,
- $ r = 0.0325 $ (the interest rate),
- $ T = 7 $ (the number of years).

Substituting in $ P(t) = 70,000 - 10,000t $, $ r = 0.0325 $, and $ T = 7 $, the formula becomes:

\[
FV = \int_0^7 (70,000 - 10,000t) e^{0.0325(7-t)} \, dt
\]

Let’s calculate this integral step by step.

### Step 1: Split the integral
\[
FV = \int_0^7 70,000 e^{0.0325(7-t)} \, dt - \int_0^7 10,000t e^{0.0325(7-t)} \, dt
\]

### Step 2: Compute the two integrals separately

The first integral:

\[
I_1 = 70,000 \int_0^7 e^{0.0325(7-t)} \, dt
\]

The second integral:

\[
I_2 = -10,000 \int_0^7 t e^{0.0325(7-t)} \, dt
\]

I'll now solve these integrals.

### Step 3: Solve integrals and compute final value
Let me complete the calculations.The future value of the account after 7 years, given the continuous stream of income and continuous compounding at 3.25%, is approximately **$285,530.20**.

Would you like more details or have any questions on how this result was obtained?

Here are five related questions to explore further:
1. How would the future value change if the interest rate were higher?
2. What if the sliding scale payment function had a different slope or initial value?
3. How does compounding frequency affect the future value in general?
4. Can we determine how much was deposited by Owen over the 7-year period?
5. How would the final amount change if the time period were extended?

**Tip**: Continuous compounding is a great tool for maximizing the return on investments over time, especially when combined with regular deposits!

Owen sells his business and receives yearly payments on a sliding scale according to the function P(t) = 70000 - 10000t. If this income is treated as a continuous stream and deposited into an account that earns an annual interest rate of 3.25% compounded continuously, how much will the account be worth after 7 years?

Learn how to calculate the future value of a sliding scale income stream deposited into an account with continuous compounding interest. The problem covers key calculus concepts such as integration, exponential growth, and continuous compounding, with an interest rate of 3.25%. Suitable for advanced high school and undergraduate students.

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