Calculus Question

Question

At a price of $8.25 per ticket, a musical theater group can fill every seat in their 1400 seat performance hall. For every additional dollar charged for admission, the number of tickets sold drops by 100. 

a) What ticket price maximizes revenue? Round your answer to the nearest cent. 

price =

b) How many seats are sold at that price? Round your answer to the nearest whole number. 

number of seats sold =

Answer 1

$11.13 per ticket maximizes profits

at that price they sell 288 tickets

for a maximum profit of 288(11.13) = $3,205.44

At price = $22.25 per ticket, no one buys any.

At price = $0 (free tickets) they can only give away 575

these two points are the x and y intercepts, if you graph the linear demand curve.

Profit maximizing point is the midpoint = approximately (288, 11.13), that's half the x intercept, half the y intercept

Maximum profit = 288 x 11.13 = $3,205.44

BUT you probably want to work this problem with some calculus

the demand curve is

P = 2225 - 2225x/575

Profit = Px = 2225x - 2225x^2/575

take the derivative and set it equal to zero

(Px)' = 2225 - 4450x/575 = 0

x = 2225(575/4450) = 575/2 = 288

profit maximizing price = 2225-2225(288)/575 = 2225/2 = $11.13

max Px = 2225(288) - 2225(288)^2/575 = 2225(144) = $3204

some rounding errors for a couple cents here and there