The monthly revenue from Monongalia County Walmart stores was ​$35.3 million in 2003. In 2016 the monthly revenue had increased to ​$104.2 million.
​(a Find the linear model for the monthly revenue of Mon.County Walmart stores​ (in millions of​ dollars) x years since 2000. 
​R(x)=
  
enter your response here
​(round coefficients to one decimal​ place)
​(b) Estimate the revenue of Mon. County walmart stores in 2029.
​$
  
enter your response heremillion
​ (round to 1 decimal​ place)
 
​(c) When will the revenue be expected to hit ​$204.9 ​million? 
In the​ year: 
  
enter your response here
​ (round to the nearest​ year)


Let's break down the problem step by step:

### Part (a): Finding the Linear Model

You are given two data points for revenue:
- In 2003, $ x = 3 $ (years since 2000), the revenue $ R(x) = 35.3 $ million.
- In 2016, $ x = 16 $ (years since 2000), the revenue $ R(x) = 104.2 $ million.

The formula for a linear equation is given by:
\[
R(x) = mx + b
\]
where $ m $ is the slope and $ b $ is the y-intercept.

#### Step 1: Calculate the slope $ m $
The slope is calculated using the formula:
\[
m = \frac{R(x_2) - R(x_1)}{x_2 - x_1}
\]
Substitute the values:
\[
m = \frac{104.2 - 35.3}{16 - 3} = \frac{68.9}{13} \approx 5.3
\]

#### Step 2: Find the y-intercept $ b $
We can use one of the points (e.g., $ x = 3, R(x) = 35.3 $) to find $ b $.
Substitute into the equation $ R(x) = mx + b $:
\[
35.3 = 5.3(3) + b
\]
\[
35.3 = 15.9 + b
\]
\[
b = 35.3 - 15.9 = 19.4
\]

Thus, the linear model is:
\[
R(x) = 5.3x + 19.4
\]

### Part (b): Estimate the Revenue in 2029

In 2029, $ x = 29 $ (since 2029 is 29 years since 2000).

Substitute $ x = 29 $ into the linear model:
\[
R(29) = 5.3(29) + 19.4 = 153.7 + 19.4 = 173.1
\]

So, the estimated revenue in 2029 is **$173.1 million**.

### Part (c): When will the revenue be expected to hit $204.9 million?

We set $ R(x) = 204.9 $ and solve for $ x $:
\[
204.9 = 5.3x + 19.4
\]
Subtract 19.4 from both sides:
\[
204.9 - 19.4 = 5.3x
\]
\[
185.5 = 5.3x
\]
Solve for $ x $:
\[
x = \frac{185.5}{5.3} \approx 35
\]

Thus, the revenue is expected to hit $204.9 million in **35 years since 2000**, which means the year will be **2035**.

### Final Answers:
- (a) The linear model is $ R(x) = 5.3x + 19.4 $
- (b) Estimated revenue in 2029: **$173.1 million**
- (c) The revenue will hit $204.9 million in the year **2035**

Would you like more details on any of these steps or additional clarification?

#### Related Questions:
1. How would the model change if the revenue increased at a different rate?
2. How does the linear model compare to an exponential model for growth?
3. What factors could affect the accuracy of this linear model over time?
4. How would the revenue be affected if the store expanded at twice the rate of increase?
5. What would be the revenue estimate in 2050 using this model?

**Tip:** Always check if the linear growth assumption is valid over long periods, as real-world factors may cause non-linear trends.

The monthly revenue from Monongalia County Walmart stores was $35.3 million in 2003. In 2016 the monthly revenue had increased to $104.2 million. (a) Find the linear model for the monthly revenue of Monongalia County Walmart stores (in millions of dollars) x years since 2000. R(x)= (round coefficients to one decimal place) (b) Estimate the revenue of Mon. County Walmart stores in 2029. $ (round to 1 decimal place) million (c) When will the revenue be expected to hit $204.9 million? In the year: (round to the nearest year)

Learn how to find the linear model for the monthly revenue of Monongalia County Walmart stores, estimate the revenue in 2029, and predict when it will hit $204.9 million. This problem covers key algebraic concepts, such as slope and y-intercept, with real-world data. Suitable for students in Grades 9-12.

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