What price per ticket should be charged in order to maximize revenue?

Question

A football team is trying to determine what price to charge for tickets. At a price of $28 per ticket, it averages 47,000 people per game. For every increase of $1 it loses 500 people.

Answer choices:

$61

$89

$28

$33

none of these

Answer 1

price = $61 to maximize revenue

points on the demand curve are (47,28) and (61,0) where (x,y) is x in thousands and y in dollars

if price is reduced by $28 to zero, 500x28= 1400 more tickets for 14+47 thousand = 61,000 if it were free.

slope of the line through the two points is -28/14 =-2

y=-2x + b

0=-2(61)+b

b = $122, at a price of $122 no one buys a ticket

y=-2x+122

revenue = xy = 2x^2 +122x

take the derivative and set = 0

(xy)' = 4x +122 = 0

x = 122/4 = 30.5

y=-2(30.5) + 122 = -61+122 = $61 for the price that maximizes revenue