To find the linear demand function, $q(p)$, we can use the two points provided in the problem that correspond to the price-demand relationship.

Step 1: Identify the points

From the problem, we have two price-demand pairs:

1. When the price is $p = 4$, the quantity demanded is $q = 3800$.
2. When the price is $p = 9$, the quantity demanded is $q = 3500$.

These points are: $(4, 3800)$ and $(9, 3500)$.

Step 2: Use the slope formula

The general form of a linear equation is: $q(p) = mp + b$ where $m$ is the slope of the line and $b$ is the intercept. The slope can be found using the formula: $m = \frac{q_2 - q_1}{p_2 - p_1}$ Using the points $(4, 3800)$ and $(9, 3500)$: $m = \frac{3500 - 3800}{9 - 4} = \frac{-300}{5} = -60$

Step 3: Solve for the intercept $b$

Now that we know the slope $m = -60$, we can use one of the points to solve for the intercept $b$. Using the point $(4, 3800)$: $q(p) = -60p + b$ Substitute $p = 4$ and $q = 3800$: $3800 = -60(4) + b$ $3800 = -240 + b$ $b = 3800 + 240 = 4040$

Step 4: Write the demand function

The linear demand function is: $q(p) = -60p + 4040$

Would you like further clarification or any more details on this?

Here are 5 related questions to consider:

1. How does the price elasticity of demand change as the price increases?
2. If the store increases the price to $7, how many shower curtains will they sell based on this model?
3. How would the demand function change if the store planned to buy more or fewer curtains at different price points?
4. Can we predict the revenue function based on this demand function?
5. How would external factors like market competition affect this demand model?

Tip: The slope in a linear demand function reflects how sensitive quantity demanded is to price changes. A steeper slope indicates more sensitivity.