The Valley High School student council is sponsoring a dance. They anticipate that 500 students will attend. The ticket price is $3.00, and they estimate that for every $0.20 increase in ticket price, 25 fewer students will attend. What ticket price will maximize the student council’s profit?

Question

anonymous · Answer

Let x be the number of groups of 25 students.

Profit = price * quantity
= (3 +0.2x) * (500 - 25x) 
= 1500  - 75x + 100x - 5x^2
= 1500 +25x - 5x^2

To find maximum profit, differentiate it.
d(profit)/dx = -10x +25
Differentiate it once more to get -10, this shows that it is indeed a maximum point since it is less than zero.

Let -10x+25 = 0
25= 10x
x= 2.5 

When x = 2.5, it is the maximum point. 
When x = 2.5 the ticket prize is, 3 +0.2(2.5) =3 + 0.5  = 3.5

Ticket prize is $3.50

smarty · Answer

Thanks, alot.!!!!

anonymous · Answer

Np. Hope you understand the workings, if you are unsure about anything, ask away. i will answer.

smarty · Answer

I got it! Work was very clear