In Gas Pump, South Dakota, there are two kinds of consumers, Buick owners and Dodge owners. Every Buick owner has a demand function for gasoline DB(p)=20-5p for p</=4 and DB(p)=0 if p>4. Every Dodge owner has a demand function DD(p)=15-3p for p</=5 and DD(p)=0 for p>5. (Quantities are measured in gallons per week and price is measured in dollars.) Suppose that Gas Pump has 150 consumers, 100 Buick owners, and 50 Dodge owners. If the price is $3, what is the total amount demanded by each individual Buick Owner? And by each individual Dodge owner?

On a graph, use blue ink to draw the demand curve representing the total demand by Buick owners. Use black ink to draw the demand curve representing total demand by Dodge owners. Use red ink to draw the market demand curve for the whole town.

To find the total demand by each consumer at the price $ p = 3 $, we need to evaluate the individual demand functions for Buick and Dodge owners at this price. Let's break down the steps:

### Step 1: Demand by Each Individual Buick Owner
The demand function for a Buick owner is:

\[
D_B(p) = 20 - 5p \quad \text{for} \quad p \leq 4, \quad D_B(p) = 0 \quad \text{if} \quad p > 4.
\]

For $ p = 3 $:

\[
D_B(3) = 20 - 5(3) = 20 - 15 = 5 \quad \text{gallons per week}.
\]

Each Buick owner demands **5 gallons** of gasoline when the price is $3.

### Step 2: Demand by Each Individual Dodge Owner
The demand function for a Dodge owner is:

\[
D_D(p) = 15 - 3p \quad \text{for} \quad p \leq 5, \quad D_D(p) = 0 \quad \text{if} \quad p > 5.
\]

For $ p = 3 $:

\[
D_D(3) = 15 - 3(3) = 15 - 9 = 6 \quad \text{gallons per week}.
\]

Each Dodge owner demands **6 gallons** of gasoline when the price is $3.

### Step 3: Graphing the Demand Curves
Next, we'll represent the demand curves graphically.

#### 1. **Buick Owners' Total Demand Curve (Blue Ink)**:
Since there are 100 Buick owners, the total demand for gasoline by all Buick owners at price $ p $ is $ 100 \times D_B(p) $. The equation for the total demand of Buick owners is:

\[
Q_B(p) = 100 \times (20 - 5p) = 2000 - 500p \quad \text{for} \quad p \leq 4.
\]

For $ p > 4 $, total demand is zero.

This curve should start at $ p = 0 $ with a demand of 2000 gallons and slope downwards at a rate of 500 gallons per dollar until it hits zero demand at $ p = 4 $.

#### 2. **Dodge Owners' Total Demand Curve (Black Ink)**:
For Dodge owners, there are 50 consumers, so the total demand is $ 50 \times D_D(p) $. The equation for the total demand of Dodge owners is:

\[
Q_D(p) = 50 \times (15 - 3p) = 750 - 150p \quad \text{for} \quad p \leq 5.
\]

For $ p > 5 $, total demand is zero.

This curve starts at $ p = 0 $ with a demand of 750 gallons and slopes downwards at a rate of 150 gallons per dollar until it hits zero demand at $ p = 5 $.

#### 3. **Market Demand Curve for the Whole Town (Red Ink)**:
The market demand is the sum of the total demand from both Buick and Dodge owners. Therefore, the market demand is:

\[
Q_{\text{market}}(p) = Q_B(p) + Q_D(p) = (2000 - 500p) + (750 - 150p) = 2750 - 650p \quad \text{for} \quad p \leq 4.
\]

For $ 4 < p \leq 5 $, only Dodge owners are demanding gas, so:

\[
Q_{\text{market}}(p) = 750 - 150p.
\]

For $ p > 5 $, no one demands gasoline, so:

\[
Q_{\text{market}}(p) = 0.
\]

This curve will start at $ p = 0 $ with a total demand of 2750 gallons and slope downwards until $ p = 4 $, after which only the Dodge owners contribute to the demand until $ p = 5 $, where total demand becomes zero.

---

Do you want me to sketch these curves for you? 

### Relative Questions:
1. How would the demand curves change if gasoline prices increased to $4?
2. What is the total amount of gasoline demanded by both Buick and Dodge owners when the price is $2?
3. What happens to the market demand if the number of Dodge owners increases?
4. How does the price elasticity of demand differ between Buick and Dodge owners?
5. How would the demand curves shift if both Buick and Dodge owners received a gasoline subsidy?

### Tip:
When graphing demand curves, remember that the slope of the curve reflects the rate at which demand decreases as price increases.

In Gas Pump, South Dakota, there are two kinds of consumers, Buick owners and Dodge owners. Every Buick owner has a demand function for gasoline DB(p)=20-5p for p</=4 and DB(p)=0 if p>4. Every Dodge owner has a demand function DD(p)=15-3p for p</=5 and DD(p)=0 for p>5. (Quantities are measured in gallons per week and price is measured in dollars.) Suppose that Gas Pump has 150 consumers, 100 Buick owners, and 50 Dodge owners. If the price is $3, what is the total amount demanded by each individual Buick Owner? And by each individual Dodge owner? On a graph, use blue ink to draw the demand curve representing the total demand by Buick owners. Use black ink to draw the demand curve representing total demand by Dodge owners. Use red ink to draw the market demand curve for the whole town.

Explore the total demand for gasoline in Gas Pump, South Dakota, with Buick and Dodge owners. Learn how to calculate the total gasoline demand when the price is $3, and visualize the demand curves using different ink colors for each consumer group. Perfect for students in Grades 10-12, this problem covers key concepts in linear demand functions and market demand curves.

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