Calculate the producer surplus (in dollars) at the unit price \( \bar{p}=18 \) for the supply equation \( p=10+2 q \). (Round your answer to the nearest cent.) dollars

To calculate the producer surplus at the unit price \( \bar{p} = 18 \) for the supply equation \( p = 10 + 2q \), we will follow these steps: 1. **Find the quantity supplied at the price \( \bar{p} \)**: We need to set the supply equation equal to the price and solve for \( q \): \[ 18 = 10 + 2q \] 2. **Calculate the producer surplus**: The producer surplus is the area above the supply curve and below the price level up to the quantity supplied. It can be calculated using the formula: \[ \text{Producer Surplus} = \frac{1}{2} \times \text{Base} \times \text{Height} \] where: - Base is the quantity supplied \( q \). - Height is the difference between the price \( \bar{p} \) and the minimum price at which the quantity is supplied (found from the supply equation). Let's perform the calculations step by step. ### Step 1: Find the quantity supplied We will solve the equation \( 18 = 10 + 2q \) for \( q \): \[ 18 - 10 = 2q \\ 8 = 2q \\ q = \frac{8}{2} = 4 \] ### Step 2: Calculate the minimum price at \( q = 4 \) Now, we will find the minimum price at which this quantity is supplied by substituting \( q = 4 \) back into the supply equation: \[ p = 10 + 2(4) = 10 + 8 = 18 \] ### Step 3: Calculate the producer surplus Since the price \( \bar{p} = 18 \) is equal to the price at which the quantity is supplied, the height of the triangle for producer surplus is: \[ \text{Height} = \bar{p} - \text{Minimum Price} = 18 - 18 = 0 \] Thus, the producer surplus is: \[ \text{Producer Surplus} = \frac{1}{2} \times q \times \text{Height} = \frac{1}{2} \times 4 \times 0 = 0 \] Therefore, the producer surplus at the unit price \( \bar{p} = 18 \) is \( \boxed{0} \) dollars.