We are given the function $S = 50,000 e^{-0.2x}$, where $S$ represents the sales and $x$ represents the number of months that have passed since the promotional campaign ended. We are tasked with finding $S$ when $x = 9$ months.

Step 1: Substitute $x = 9$ into the equation.

$S = 50,000 e^{-0.2 \times 9}$

Simplify the exponent:

$S = 50,000 e^{-1.8}$

Step 2: Approximate $e^{-1.8}$.

Using a calculator or approximation:

$e^{-1.8} \approx 0.165298$

Step 3: Multiply by 50,000.

Now calculate the sales:

$S = 50,000 \times 0.165298 = 8,264.90$

Final Answer:

The predicted sales after 9 months are approximately $8,264.90.

Would you like more details on any step, or have any questions?

Relative Questions:

1. What does the function suggest about the long-term sales trend?
2. How does the decay constant, $-0.2$, affect the sales over time?
3. What are some methods to calculate the value of $e^{-1.8}$ without a calculator?
4. Can the function be modified to model a different product with a slower decay rate?
5. What would sales be after 12 months using the same model?

Tip:

When dealing with exponential decay, always check the units of the decay constant to understand the time scale of the process.