Find maximized revenue

Question

An airline finds that if it prices a cross-country ticket at $300, it will sell 200 tickets per day. It estimates that each $20 price reduction will result in 40 more tickets sold per day. Find the ticket price (and the number of tickets sold) that will maximize the airline's revenue.

Answer 1

let x=the number of $20 price reductions and the (same) number of 40-ticket increases

(300-20x)(200+40x)=0

60,000-4000x+12,000x-800x²=0

60,000+8000x-800x²=0

use -b/2a

-8000/-1600=5

x=5 $20 reductions and 5 40-ticket increases

$300-$100=$200

200+200=400

$200*400=$80,000 maximum revenue

if there were 4 reductions...

$300-$80=$220, 200+160=360, $220*360=$79,200

if there were 6 reductions...

$300-$120=$180, 200+240=440, $180*440=$79,200

so 5 reductions is the answer; $200 price and 400 ticket sales