Maximizing Revenues

Date: 07/29/2002 at 22:28:05
From: AK
Subject: Max/Min Problem

The Boston Red Sox general manager has a dilemma. His team is playing 
well and will likely make the playoffs. He wants to charge more for 
tickets during the playoffs to increase revenue for next year to spend 
on free agents. The team did a survey of fans and found out that for 
every $1.00 increase in ticket price, 200 fewer fans will attend a 
single game. What should the Sox charge to maximize revenue if their 
average ticket price is $18.00 and the average attendance is 21,000 
per game? What will the maximum revenue be?

Date: 07/30/2002 at 00:12:39
From: Doctor Ian
Subject: Re: Max/Min Problem

Hi AK,

If he doesn't do anything, he'll end up with 

  21,000 * 18 = 378,000

dollars, right? Suppose he adds $1 to the ticket price. He'll end up 
with 

  (21,000 - 200) * (18 + 1) = 395,200

dollars. The simplest way to proceed would be to keep adding $1 and 
dropping 200 tickets sold until the result starts to go down instead 
of up. 

Or, you could generalize the situation as a function of the number of 
dollars added to the ticket price, 
  
  f(x) = (21,000 - 200*x) * (18 + x) 

take the derivative with respect to x, and see where the derivative is 
equal to zero, which will tell you where the function reaches its 
maximum value. 

Does that make sense?  Can you take it from here?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/30/2002 at 08:46:33
From: AK
Subject: Thank you (Max/Min Problem)

Hi, 

Thank you Dr. Ian.  That was very clearly explained!

Thanks again.