A large department store is prepared to buy 3,800 of your tie-dye shower curtains per month for $4 each, but only 3,500 shower curtains per month for $9 each. What is the linear demand function for your tie-dye shower curtains?
q(p) =

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.

A large department store is prepared to buy 3,800 of your tie-dye shower curtains per month for $4 each, but only 3,500 shower curtains per month for $9 each. What is the linear demand function for your tie-dye shower curtains? q(p) =

Learn how to find the linear demand function for tie-dye shower curtains based on the price-demand relationship. This problem involves algebraic concepts, including calculating the slope and forming a linear equation. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation