An auditorium has 1600 seats. At a ticket price of $6, the owners can sell 1200 tickets. For each 25 cent increase in price the tickets sold reduces by 25. What price should the owners set to maximize the income?

Answer 1

let the number of 25 cent increases by x
income = ticket price x number of tickets sold

income = (6+.25x)(1200-25x)
= 7200 - 150x + 300x - 6.25x^2
= -6.25x^2 150x + 7200

d(income)/dx = -12.5x + 150 = 0 for a max of income
12.5x = 150
x = 12

so the price for max income should be 6 + .25(12) = $9.00