https://file.aiquickdraw.com/mathbot/file/17302227641701u4bx1cz.jpg

Let’s go through the problem step-by-step.

### Given Model
The sales model is:
\[
S = \frac{500,000}{1 + 0.1 e^{kt}}
\]
where:
- \( S \) is the number of units sold,
- \( t \) is the number of years since 2010 (thus, \( t = 0 \) represents the year 2010),
- We know that in 2013, 300,000 units were sold (i.e., \( S = 300,000 \) when \( t = 3 \)).

### Problem Steps
1. **Part (a): Estimating Sales in 2018 from the Graph**  
   Look at the graph where \( t = 8 \) (since 2018 is 8 years after 2010) and read off the value of \( S \). 

2. **Part (b): Solving for \( k \)**
   Use the given sales data from 2013 to solve for \( k \). Substitute \( t = 3 \) and \( S = 300,000 \) into the model:
   \[
   300,000 = \frac{500,000}{1 + 0.1 e^{3k}}
   \]
   Solve this equation for \( k \).

3. **Part (c): Estimating Sales in 2018 Using the Model**
   With the value of \( k \) from part (b), substitute \( t = 8 \) into the model to find the algebraic solution for sales in 2018.

Let me work out the equations and calculations for parts (b) and (c).Here are the solutions for each part:

1. **Part (a)**: To estimate the sales in 2018 from the graph, locate \( t = 8 \) and approximate \( S \) visually. (From the graph, it appears to be around **30,000 units**.)

2. **Part (b)**: Solving for \( k \) using the data from 2013, we find:
   \[
   k \approx 0.632
   \]

3. **Part (c)**: Using the model with the calculated \( k \) value, the estimated sales in 2018 (when \( t = 8 \)) are:
   \[
   S \approx 29,863 \text{ units}
   \]

### Comparison:
The algebraic result (29,863 units) is close to the graphical estimate (about 30,000 units), so we would select **“The algebraic result is similar to the graphical result.”**

Would you like further clarification on any part?

---

### Related Questions:
1. How would changes in \( k \) affect the model's predictions?
2. What other methods can be used to verify the calculated \( k \) value?
3. How can this model be adjusted for different starting sales values?
4. What assumptions are made about advertising's impact in this model?
5. Can this type of model be used for projecting future sales if advertising is resumed?

### Tip:
When solving for unknowns in equations, use known data points to simplify and isolate variables for accurate results.

After discontinuing all advertising for a tool kit in 2010, the manufacturer noted that sales began to drop according to the model:

S = 500,000 / (1 + 0.1e^(kt))

where S represents the number of units sold and t represents the year, with t = 0 corresponding to 2010. In 2013, 300,000 units were sold. Estimate sales in 2018, solve for k, and estimate sales in 2018 using the model.

This problem explores estimating sales for a discontinued tool kit using an exponential decay model. Given that sales were 300,000 units in 2013, calculate the decay constant 'k' and estimate sales in 2018. Ideal for students in Grades 10-12, this solution covers exponential equations and graphical analysis.

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