A theater plays attendance depends on the number of positive reviews of plays per season and on the price of its tickets. The demand function it faces is Q = N(20 – p), where Q is the number of tickets (in hundred thousands) sold per year, p is the price per ticket, and N is the fraction plays with positive reviews. Hiring better directors can increase the number of plays receiving positive reviews. If the theater spends C million pounds on directors, the fraction of its reviews being positive will be equal to (0.7 – 1/C). Over the relevant range, the marginal cost of selling an extra ticket is zero. i. Write an expression for the theater's profits as a function of ticket price and expenditure on directors. ii. Find the ticket price that maximizes revenue. iii. Find the profit-maximizing expenditure on director and the profit-maximizing fraction of plays with a positive review.

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