Final answer:
The video store expects to make an annual profit of $4.75 on an average customer using optimal two-part pricing.
Step-by-step explanation:
The annual profit that the video store expects to make on an average customer using optimal two-part pricing can be calculated by subtracting the total marginal cost from the total revenue. To find the total marginal cost, we need to integrate the marginal cost function with respect to the number of rentals, which gives us 0.5q. The total revenue can be found by integrating the demand function with respect to the number of rentals, which gives us the revenue function R(q) = pq - 0.5q^2. To find the profit, we subtract the total marginal cost from the total revenue:
Profit = R(q) - 0.5q
Substituting the demand function q = 7 - 2p, we can solve for p:
Profit = (7 - 2p)p - 0.5q
Profit = 7p - 2p^2 - 0.5(7 - 2p)
Profit = 7p - 2p^2 - 3.5 + p
Profit = -2p^2 + 6p - 3.5
To maximize profit, we can take the derivative of the profit function with respect to p and set it equal to zero:
d(Profit)/dp = -4p + 6 = 0
Solving for p, we find p = 1.5. Substituting this value back into the profit function, we can find the maximum profit:
Profit = -2(1.5)^2 + 6(1.5) - 3.5 = $4.75