Optimizing Ticket Prices for Maximum Revenue

[jimen113]

[SOLVED] Optimization problem

Homework Statement

A baseball team plays in the stadium that holds 60000 spectators. With the ticket price at 8 the average attendence has been 24000. When the price dropped to 7, the average attendence rose to 30,000.
a) find the demand function p(x), where x is the number of spectators (assume p(x) is linear p(x)=
b) How should the ticket price be set to maximize revenue?

Homework Equations

The Attempt at a Solution

a) I figured out that the demand function was: 8-(1/6000)(x-24000)
b) I tried to use the revenue formula:; by taking the derivative of the demand function which is: 12-(12,000x/6000^{}2)
and then I set that equal to zero and tried to solve for X, which is: 36,000
So, now I don't know how to use the above information to calculate the price of the ticket. any suggestions?

 

[Tedjn]

So x = 36,000 is a critical number of the revenue function r(x) = xp(x) = 12x - x^2 / 6000. What does that tell you about r(x) at x = 36,000? What does that tell you about p(x)?

 

[jimen113]

I plugged in the x=36,000 into the demand function and I got $6.00 as the answer, I verified it and it is correct.
Thanks, I appreciate your help.