Maximize Revenue from ticket sales word problem

[hvidales]

Homework Statement

Homework Statement [/b]

A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket prices set at $12, average attendance at a game has been 11,000. A market survey indicates that for each dollar the ticket price is lowered, average attendance will increase by 1,000. How should the owners of the team set the ticket price to maximize their revenue from tickets sales?

Homework Equations

R(x)=xp(x) where p is called the demand function(or price function)[x units are sold and the price per unit is p(x)] or P(x) =R(x)-C(x) where x units are sold, C(x) is the cost function and P(x) is the profit.

The Attempt at a Solution

I am stuck on how to set up the problem. I used to R(x)=xp(x) formula and so far I have come up with R(x) = (12-x)(11,000+ 1000x) but I am not sure if that is correct. I was also thinking that this problem is a linear function and perhaps I can some how connect it to the point slope formula and the slope formula. Any help would be nice. Thanks in advance!

 

[CompuChip]

If the problem were linear, then R(x) would be something like ax, and the optimal solution would be x = 0 or x = infinity.

What you did looks good: if x is the amount the price is lowered, then average attendance is (11000 + 1000x). If you work out the brackets, you will find that R(x) is quadratic, so it will have a clear minimum or maximum.