You can sell 150 pet chias per week if they are marked at $1 each, but only 120 each week if they are marked at $2/chia. Your chia supplier is prepared to sell you 30 chias each week if they are marked at $1 per chia, and 100 each week if they are marked at $2 per chia.
(a)
Write down the associated linear demand and supply functions.
demand function	q(p)	=

supply function	q(p)	=

(b)
At what price (in dollars) should the chias be marked so that there is neither a surplus nor a shortage of chias?

### Part (a): Linear Demand and Supply Functions

We are given two points for the demand and supply functions, which can help us establish the linear relationships.

#### **Demand Function:**
- At a price of $ p = 1 $, the quantity demanded is 150 chias per week.
- At a price of $ p = 2 $, the quantity demanded is 120 chias per week.

The general form of a linear demand function is:
\[
q(p) = mp + b
\]
Where $ m $ is the slope and $ b $ is the y-intercept (demand when $ p = 0 $).

The slope is calculated by the change in quantity divided by the change in price:
\[
m = \frac{\Delta q}{\Delta p} = \frac{120 - 150}{2 - 1} = \frac{-30}{1} = -30
\]

Now, using one point to find $ b $, let's use $ p = 1 $ and $ q = 150 $:
\[
150 = -30(1) + b
\]
\[
b = 150 + 30 = 180
\]

Thus, the demand function is:
\[
q(p) = -30p + 180
\]

#### **Supply Function:**
- At a price of $ p = 1 $, the quantity supplied is 30 chias per week.
- At a price of $ p = 2 $, the quantity supplied is 100 chias per week.

The general form of the supply function is similar:
\[
q(p) = mp + b
\]

The slope is:
\[
m = \frac{\Delta q}{\Delta p} = \frac{100 - 30}{2 - 1} = \frac{70}{1} = 70
\]

Now, using the point $ p = 1 $ and $ q = 30 $:
\[
30 = 70(1) + b
\]
\[
b = 30 - 70 = -40
\]

Thus, the supply function is:
\[
q(p) = 70p - 40
\]

### Part (b): Finding the Equilibrium Price

To find the equilibrium price, we set the demand equal to the supply, i.e., $ q_{\text{demand}}(p) = q_{\text{supply}}(p) $:
\[
-30p + 180 = 70p - 40
\]

Now solve for $ p $:
\[
180 + 40 = 70p + 30p
\]
\[
220 = 100p
\]
\[
p = \frac{220}{100} = 2.20
\]

So, the equilibrium price is **$2.20** per chia.

### Final Answer:
- **Demand function**: $ q(p) = -30p + 180 $
- **Supply function**: $ q(p) = 70p - 40 $
- **Equilibrium price**: $2.20

Would you like further details or have any questions? Here are 5 related questions to explore the concepts more:

1. What happens to the equilibrium quantity when the demand or supply shifts?
2. How would a government-imposed price ceiling below the equilibrium price affect the chia market?
3. What is the equilibrium quantity of chias at the price of $2.20?
4. How does elasticity of demand or supply impact market equilibrium?
5. How would an increase in supplier costs affect the supply function?

**Tip**: Equilibrium is where supply and demand meet, but external factors can cause shifts leading to either surplus or shortage.

You can sell 150 pet chias per week if they are marked at $1 each, but only 120 each week if they are marked at $2/chia. Your chia supplier is prepared to sell you 30 chias each week if they are marked at $1 per chia, and 100 each week if they are marked at $2 per chia.
(a) Write down the associated linear demand and supply functions.
(b) At what price (in dollars) should the chias be marked so that there is neither a surplus nor a shortage of chias?

Solve a real-world problem involving linear demand and supply functions. Learn how to calculate demand and supply slopes, form equations, and find the equilibrium price where there is neither surplus nor shortage. Ideal for 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