Question 13 Calculate the consumer surplus (in dollars) at the unit price $\bar{p}=55$ for the demand equation $p=420 e^{-2 q}$. Round your answer to the nearest cent.
Solution
Step 1 ：Given the demand function \(p = 420e^{-2q}\), where \(p\) is the price per unit and \(q\) is the quantity.
Step 2 ：We are asked to find the consumer surplus at the unit price \(\bar{p} = 55\).
Step 3 ：First, we need to find the quantity demanded at this price. We do this by setting \(p = 55\) in the demand function and solving for \(q\).
Step 4 ：The solution to the equation gives the quantity demanded as \(q = 1.5\).
Step 5 ：The consumer surplus is the area between the demand curve and the price level up to the quantity demanded at the unit price. It is calculated as the integral of the demand function from 0 to the quantity demanded, minus the total amount of money consumers actually paid, which is the unit price times the quantity demanded.
Step 6 ：The consumer surplus is calculated as \(315.75\).
Step 7 ：Final Answer: The consumer surplus at the unit price \(\bar{p} = 55\) is \(\boxed{315.75}\).
