at time t in years, the sales are s(t) = 50e^-t thousands of dollars per year. After two years, all money is invested into bonds that pay interest at a 6.1% rate per year compounded continuously. Calculate the total value of sales over the two year period

To calculate the total value of sales over the two-year period, we need to find the integral of the sales function $ s(t) $ from $ t = 0 $ to $ t = 2 $. The sales function is given by:

\[
s(t) = 50e^{-t} \quad \text{(in thousands of dollars per year)}
\]

The total sales value over the period is the definite integral of $ s(t) $ from 0 to 2.

\[
\text{Total Sales} = \int_{0}^{2} s(t) \, dt
\]

Let's calculate this integral.The total value of sales over the two-year period is approximately \$43,233.24 (in thousands of dollars), which translates to \$43,233,240.

Would you like further details on this calculation or have any other questions?

Here are some related questions:
1. What is the total sales value if the time period extends to 3 years?
2. How does the sales value change if the decay rate is different, say $ e^{-2t} $?
3. How can the future value of these sales be calculated if invested at a different interest rate?
4. What is the significance of the continuous decay factor in the sales function?
5. How would the total sales value change if the initial sales rate was 100 instead of 50?
6. What would be the effect on the total sales if the decay rate was halved?
7. How can we model a scenario where sales increase exponentially instead?
8. What are the practical implications of declining sales and how can businesses mitigate this?

**Tip:** When dealing with continuously compounded rates or decaying functions, integrating over time provides a more accurate picture of the accumulated values or costs.

Learn how to calculate the total sales value over a two-year period using an exponential decay function. The problem involves integrating the sales function s(t) = 50e^{-t} from t = 0 to t = 2. The total sales value is approximately $43,233.24 (in thousands of dollars), which translates to $43,233,240. Explore related questions such as extending the time period, changing the decay rate, and calculating future values with different interest rates.

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