Please help with this math question

Question

54,000 spectators with the tickets price at $11 the average attendance has been 23,000. When the price dropped to eight dollars the average attendance rules to 27,000. Assume the attendance is linearly related to ticket price. What ticket price would maximize revenue?

Answer 1

Let find function for price: we have two cases: (23000, 11) and (27000, 8). So slope m = (8 - 11)/(27000 - 23000) = - 3/4000x. So price equation p(x) = 8 - 3/4000(x - 27000) = 28.25 - 0.00075x.

Then revenu function R(x) = xp(x) = 28.25x - 0.00075x². To get maximum this function let find mmerginal revenue: r'(x) = 28.25 - 0.0015x = 0. So x = 28.25/0.0015. x = 56500/3.

So this prse is p_max = 28.25 - 0.00075·56500/3 = $14.13