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how do I solve this for business calculus ?

The problem you've shared involves the concept of **exponential decay**, which is commonly seen in business calculus, especially in sales models after a campaign ends. Let's break down the steps to solve both parts.

### Given Equation:
\[
S = 80,000 e^{-0.1x}
\]
where:
- $ S $ is the weekly sales in dollars.
- $ x $ is the number of weeks that have passed since the campaign ended.
- $ e $ is the base of the natural logarithm.

### Part (a): What will sales be 6 weeks after the end of the campaign?
You're given $ x = 6 $. Simply substitute 6 into the equation to find $ S $.

\[
S = 80,000 e^{-0.1(6)}
\]
\[
S = 80,000 e^{-0.6}
\]
Now calculate $ e^{-0.6} $. Using a calculator:

\[
e^{-0.6} \approx 0.5488
\]

Now, multiply by 80,000:

\[
S \approx 80,000 \times 0.5488 = 43,904.93
\]

So, the sales 6 weeks after the campaign end are approximately **$43,904.93**.

### Part (b): How many weeks will pass before sales drop below $16,000?
Here you're given $ S = 16,000 $, and you need to solve for $ x $.

Start by setting the equation equal to 16,000:

\[
16,000 = 80,000 e^{-0.1x}
\]

To isolate $ e^{-0.1x} $, divide both sides by 80,000:

\[
\frac{16,000}{80,000} = e^{-0.1x}
\]
\[
0.2 = e^{-0.1x}
\]

Now take the natural logarithm of both sides to eliminate the $ e $:

\[
\ln(0.2) = -0.1x
\]
\[
x = \frac{\ln(0.2)}{-0.1}
\]

Use a calculator to find $ \ln(0.2) $:

\[
\ln(0.2) \approx -1.6094
\]

Now solve for $ x $:

\[
x = \frac{-1.6094}{-0.1} = 16.094
\]

Rounding down to the nearest week, you get $ x = 16 $.

So, it will take approximately **16 weeks** for the sales to drop below $16,000.

---

### Do you have any questions or would you like further explanation on these steps?

Here are 5 related questions to expand your understanding:

1. How can you interpret exponential decay in the context of business forecasting?
2. What happens to sales over a long period based on this model?
3. How would you adjust the model if the decay rate changes?
4. Can this model be used for other industries or processes that exhibit exponential decay?
5. How does the natural logarithm help solve for the number of weeks in Part (b)?

**Tip:** Understanding the behavior of exponential functions, especially with respect to time, is crucial in both finance and business modeling.

The sales decay for a product is given by S = 80,000e^(-0.1x). What will sales be 6 weeks after the end of the campaign? How many weeks will pass before sales drop below $16,000?

Learn how to solve sales decay using the exponential model S = 80,000e^(-0.1x). Find weekly sales after a campaign and determine the time until sales drop below $16,000. Step-by-step solution and explanation, suitable for Grades 10-12.

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