Final answer:
To determine the profit-maximizing quantity sold to students, the marginal revenue (MR) must be set equal to the marginal cost (MC). Calculus and algebra are then used to solve for the ticket price and quantity. The demand function for student tickets is qs = 10,000 - 100p, with a constant marginal and average total cost of $10.
Step-by-step explanation:
To calculate the profit-maximizing quantity of tickets sold to students, we utilize the demand function for student tickets qs = 10,000 - 100p. Here, p represents the price of the tickets and qs is the quantity sold to students. We also know that the marginal cost (MC) and average total cost (ATC) of seating an additional spectator is constant at $10.
Since we are trying to maximize profits, we need to set the marginal revenue (MR) equal to the marginal cost (MC). Marginal revenue can be found by the derivative of the total revenue function, which is the price p times the quantity of tickets sold qs. The total revenue function TR for students is p times (10,000 - 100p), and by taking the derivative, we get the MR function for students.
To find the profit-maximizing price that corresponds to the MR = MC, and then use that price to determine the quantity sold to students qs, we need to do some calculus and algebra. The goal is to find the value of qs that will maximize profits. However, without being able to solve the derivative function here, we cannot directly provide the quantity. Still, the approach would involve setting MR equal to $10, finding the price, and then plugging it back into the demand function for qs.