Optimization question

Question

A baseball team plays in a stadium that holds 64000 spectators. With the ticket price at $9 the average attendance has been 27000. When the price dropped to $8, the average attendance rose to 32000.

 

a)Find the demand function p(x) , where x is the number of the spectators. (Assume that p(x) is linear.)

 

b)How should ticket prices be set to maximize revenue?

Answer 1

Let x = the ticket cost reduction in $.  The ticket price function will be:

 

Price = $9 - x

 

The demand function (number of spectators) is:

 

p(x) = 27,000 + 5000x

 

The revenue is:

 

R(x) = (Demand)(Price)

R(x) = (27,000 + 5000x)(9 - x) = -5000x² + 180,000x + 243,000

 

To find the optimal revenue, take the derivative of R(x) wrt x, set it to zero, and solve for x.  Price = $9 - x.  Answer is $7.20.