Assume that the demand curve D(p) given below is the market demand for widgets:

Q=D(p)=2606−21p
, p > 0

Let the market supply of widgets be given by:

Q=S(p)=−4+8p
, p > 0

where p is the price and Q is the quantity. The functions D(p) and S(p) give the number of widgets demanded and supplied at a given price.

The problem involves determining the equilibrium price and quantity for the market of widgets, where the demand and supply are represented as functions of price.

We are given:

- **Demand function**: \( D(p) = 2606 - 21p \), where \( p > 0 \)
- **Supply function**: \( S(p) = -4 + 8p \), where \( p > 0 \)

To find the **equilibrium price** and **equilibrium quantity**, we set the demand equal to the supply because, at equilibrium, the quantity demanded equals the quantity supplied:

\[
D(p) = S(p)
\]

### Step 1: Set the demand equal to supply.

\[
2606 - 21p = -4 + 8p
\]

### Step 2: Solve for \( p \) (the equilibrium price).

Move all terms involving \( p \) to one side and constants to the other:

\[
2606 + 4 = 21p + 8p
\]

\[
2610 = 29p
\]

Now, solve for \( p \):

\[
p = \frac{2610}{29} \approx 90
\]

So, the **equilibrium price** is approximately \( p = 90 \).

### Step 3: Find the equilibrium quantity.

Substitute \( p = 90 \) into either the demand function \( D(p) \) or the supply function \( S(p) \). Let's use the demand function:

\[
Q = D(90) = 2606 - 21(90)
\]

\[
Q = 2606 - 1890 = 716
\]

Thus, the **equilibrium quantity** is \( Q = 716 \).

### Conclusion:

- **Equilibrium price**: \( p = 90 \)
- **Equilibrium quantity**: \( Q = 716 \)

Would you like further details or have any questions?

---

Here are 5 related questions for further exploration:

1. What happens to the equilibrium price and quantity if the demand curve shifts to \( D(p) = 2706 - 21p \)?
2. How does a change in the supply function, say to \( S(p) = 4 + 8p \), affect the equilibrium?
3. What is the price elasticity of demand at the equilibrium price?
4. How would the market react if a price ceiling is imposed below the equilibrium price?
5. Can you graph the demand and supply functions to visualize the equilibrium?

**Tip**: When solving for equilibrium, remember that changes in either demand or supply will shift the equilibrium price and quantity accordingly!

Assume that the demand curve D(p) given below is the market demand for widgets: Q=D(p)=2606−21p, p > 0. Let the market supply of widgets be given by: Q=S(p)=-4+8p, p > 0, where p is the price and Q is the quantity. The functions D(p) and S(p) give the number of widgets demanded and supplied at a given price. Find the equilibrium price and quantity.

Discover how to find the equilibrium price and quantity for widgets using linear demand and supply functions. This problem focuses on solving for price and quantity using algebraic methods. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation