Question

A theater seats 200 people and charges $10 for a ticket. At this price, all the tickets can be sold. A survey indicates that if the ticket price is increased by a peso, the number of tickets sold will decrease by 10. What price will yield the greatest revenue?

Answer 1

number of increases of one unit of money (pesos ? or dollars ?) be n

price = 10+n
number sold = 200-10n

revenue
= R = (10+n)(200-10n)
= 2000 + 100n - 10n^2
completing the square
= -10(n^2 - 10n + 25 -25) + 2000
= -10(n-5)^2 + 2250

vertex is (5,2250)

price should be 15
max revenue is 2250

or , by Calculus

dR/dn = 100 - 20n = 0 for a max of R
20n = 100
n = 5 , as before