The picturesque village of Horsehead, Massachusetts, lies on a bay that is inhabited by the delectable crustacean, homarus americanus, also known as the lobster. The town council of Horsehead issues permits to trap lobsters and is trying to determine how many permits to issue. The economics of the situation is this:
1. It costs $2,000 dollars a month to operate a lobster boat.
2. If there are x boats operating in Horsehead Bay, the total revenue from the lobster catch per month will be f(x) = $1, 000(10x − x2).
(a) In the graph below, plot the curves for the average product, AP(x) = f(x)/x, and the marginal product, MP(x) = 10, 000 − 2, 000x. In the same graph, plot the line indicating the cost of operating a boat.
(b) If the permits are free of charge, how many boats will trap lobsters in Horsehead, Massachusetts?
(c) What number of boats maximizes total profits?
(d) If Horsehead, Massachusetts, wants to restrict the number of boats to the number that maximizes total profits, how much should it charge per month for a lobstering permit?