Applied Optimization Problems

Question

A baseball team plays in a stadium that holds 66000 spectators. With the ticket price at $8 the average attendance has been 30000. When the price dropped to $7, the average attendance rose to 33000. Assume that attendance is linearly related to ticket price.

What ticket price would maximize revenue?

Answer 1

Let x = ticket price and y = attendance

(8, 30000) and (7, 33000) are points on a line

Slope = (33000-30000) / (7-8) = -3000

Equation of line: y - 30000 = -3000(x - 8)

So y = -3000x + 54000 When y = 66000, x = 4 and when y = 0, x = 18

Revenue if price is x: R(x) = (price per ticket)(number of tickets sold) = xy = x(-3000x + 54000)

R(x) = -3000x² + 54000x, where 4 ≤ x ≤ 18

R'(x) = -6000x + 54000

R'(x) = 0 when x = 9

Maximum revenue when x = $9 per ticket.