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

Q=D(p)=1496−12p𝑄=𝐷(𝑝)=1496-12𝑝, p > 0

Let the market supply of widgets be given by:

Q=S(p)=−4+8p𝑄=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. What is the producer surplus at equilibrium?

To find the **producer surplus** at equilibrium, we first need to determine the **equilibrium price** and **quantity** where supply equals demand. This happens when \( D(p) = S(p) \).

### Step 1: Set \( D(p) = S(p) \)
Given:
\[
D(p) = 1496 - 12p
\]
\[
S(p) = -4 + 8p
\]
At equilibrium:
\[
1496 - 12p = -4 + 8p
\]
### Step 2: Solve for \( p \) (equilibrium price)
First, add \( 12p \) to both sides:
\[
1496 = -4 + 20p
\]
Next, add 4 to both sides:
\[
1500 = 20p
\]
Now, divide both sides by 20 to find \( p \):
\[
p = \frac{1500}{20} = 75
\]

Thus, the equilibrium price is \( p = 75 \).

### Step 3: Find the equilibrium quantity \( Q \)
Substitute \( p = 75 \) into either the demand or supply function. Let's use the supply function \( S(p) = -4 + 8p \):

\[
Q = -4 + 8(75) = -4 + 600 = 596
\]

Thus, the equilibrium quantity is \( Q = 596 \).

### Step 4: Calculate the Producer Surplus
The producer surplus is the area above the supply curve and below the equilibrium price.

1. The equilibrium price is \( p = 75 \).
2. To find the price where the supply curve starts (where \( Q = 0 \)), set \( S(p) = 0 \):
   \[
   0 = -4 + 8p
   \]
   \[
   8p = 4 \quad \Rightarrow \quad p = \frac{4}{8} = 0.5
   \]

Thus, the supply curve starts at \( p = 0.5 \).

3. The producer surplus is the area of the triangle with base \( 75 - 0.5 = 74.5 \) and height \( 596 \) (the equilibrium quantity).

The formula for the area of a triangle is:
\[
\text{Producer Surplus} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the values:
\[
\text{Producer Surplus} = \frac{1}{2} \times 74.5 \times 596 = 22,226.5
\]

### Final Answer:
The producer surplus at equilibrium is **22,226.5**.

---

Would you like further details on any step or have any questions? 

Here are 5 related questions to deepen understanding:
1. How is consumer surplus different from producer surplus?
2. What happens if the supply function changes?
3. How do we calculate total surplus in the market?
4. What is the significance of the equilibrium price in a market?
5. How would producer surplus change if demand increases?

**Tip:** The area under the supply curve and above the price line can also be used to calculate the producer surplus geometrically if the function is nonlinear.

Assume that the demand curve D(p) given below is the market demand for widgets: Q=D(p)=1496−12p, 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. What is the producer surplus at equilibrium?

Learn how to calculate the producer surplus at equilibrium for a market with given supply and demand functions. This problem explores algebraic concepts like equilibrium conditions and applies the area of a triangle formula to find the surplus. Ideal for students in Grades 11-12.

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