The function 
S = 50,000e−0.2x
 models sales decay for a given product. The variable S represents the monthly sales and the variable x represents the number of months that have passed since the promotional campaign ended.
If 9 months have passed since the promotional campaign ended, then 
x =   
Correct: Your answer is correct.
 can be substituted into the given function to compute the sales, S, for this product at this point in time.
Correct: Your answer is correct.

Since S is measured in dollars, round the answer to the nearest cent (hundredth). Nine months after the promotional campaign ends, the sales are predicted to be $

To solve the problem, we start with the given function:

\[
S = 50,000e^{-0.2x}
\]

We are asked to find the sales, $ S $, when $ x = 9 $ months. Let's substitute $ x = 9 $ into the equation:

\[
S = 50,000e^{-0.2 \cdot 9}
\]

Now, we need to perform the calculation:

1. Calculate the exponent:
   \[
   -0.2 \cdot 9 = -1.8
   \]
   
2. Substitute back:
   \[
   S = 50,000e^{-1.8}
   \]

3. Compute $ e^{-1.8} $ (using the approximate value $ e \approx 2.71828 $):
   \[
   e^{-1.8} \approx 0.1653
   \]

4. Multiply by 50,000:
   \[
   S \approx 50,000 \times 0.1653 \approx 8,265
   \]

Therefore, the predicted sales after 9 months are approximately $ S \approx 8,265.00 $.

Let me know if you want more details or have any questions!

Here are some related questions:
1. What is the half-life of sales decay based on this model?
2. How would the sales change if the decay rate was increased to -0.3 instead of -0.2?
3. What would the sales be 12 months after the campaign ends?
4. How can we determine the time when sales drop below $1,000?
5. What is the initial sales value when $ x = 0 $?

**Tip:** When dealing with exponential decay, the base $ e $ allows for continuous modeling of growth or decay, making it particularly useful in finance and sales predictions.

The function S = 50,000e−0.2x models sales decay for a given product. The variable S represents the monthly sales and the variable x represents the number of months that have passed since the promotional campaign ended. If 9 months have passed since the promotional campaign ended, calculate the sales S for this product at this point in time. Round the answer to the nearest cent.

Learn how to calculate the sales decay for a product using the exponential decay model S = 50,000e^{-0.2x}. This example solves for sales 9 months after the campaign ends, predicting sales to be approximately $8,265. Explore key mathematical concepts like exponential decay and sales modeling. Suitable for Grades 10-12.

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