Solve Calculus Problem: Find Max Profit of The Sound of Music Tickets

[the_morbidus]

Homework Statement

Currently 2000 people attend performances of The Sound of Music if tickets cost $40. Expenses are $8 per person in attendance for each performance. For each $2 decrease in the ticket price, 200 more people attend. Calculate the ticket price that produces maximum profit.

Homework Equations

don't really know of relevant equations besides derivatives.

The Attempt at a Solution

So I'm struggling to solve this one, I've been trying to use another similar problem to help me solve this and so i have come up with 2 functions for it.

2000+200x for how much people will come in
40-2x for the minimum price for the fare
now i don't know where to put $8 and well i just can't figure out the main function so i can come up with a derivative and solve for x.

 

[Borek]

What are expenses when n people attend?

What is profit?

 

[the_morbidus]

well i think the expences will be 8 dollars per person so 8(2000+200x)
the profit before expenses will be... (40-2x)(2000+200x) ??
so overall profit will be mmm P = (40-2x)(2000+200x)-8(2000+200x) ?

man, optimization problems are so tricky...but i can't give up, worth a lot of marks hahaha.

 

[Borek]

I can be missing something, but so far looks OK to me.