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4. (24 points) A lake is used for fishing. The total number of
fish caught in the lake is given by T P = 100B − B2 , where B is
the number of boats in the lake. The number of fish caught by each
boat is AP = 100 − B. The cost of using each boat for fishing is
40. The market price of each fish is 2.

(a) What is the number of boats B that maximizes the total
profit from fishing?

(b) Assume that access to the lake is unrestricted. How many
boats will use the lake?

(c) Suppose that every boat must pay an access fee F in order to
use the lake. What access fee F would reduce the number of boats in
the lake to the level in part (a)?

(d) Assume that access to the lake is unrestricted. Suppose that
every boat must pay a tax T on each fish caught in the lake. What
tax T would reduce the number of boats in the lake to the level in
part (a)?