What ticket price would maximize revenue?

Question

A baseball team plays in a stadium that holds 64000 spectators. With the ticket price at $8 the average attendance has been 28000. When the price dropped to $7, the average attendance rose to 32000. Assume that attendance is linearly related to ticket price.

What ticket price would maximize revenue? 

Answer 1

From the problem description we know that a one dollar decrease in ticket price results in a 4000 person increase in attendance. We can therefore write:

A = Ao - T * 4000

Where A is the attendance, Ao is a constant, and T is the ticket price. From the data given we have

28000 = Ao - 8 * 4000

32000 = Ao - 7 * 4000

or

Ao = 60000.

We also know that revenue R is

R = A * T = (Ao - T * 4000) * T = Ao * T - 4000 * T^2

To find the maximu we take the first derivative of R wrt T and set it to zero

dR/dT = Ao - 8000 * T = 0

So the maximum is at a ticket price T = Ao / 8000 = $7.50