Maximum revenue?

Question

A club currently has 300 members who pay $400 per month for membership dues. The club's board members want to increase monthly revenue by lowering the monthly dues in hopes of attracting new members. A market research study has shown that for each $1 decrease in the monthly membership price, an additional 2 people will join the club. What price should the club charge to maximize revenue? What is the maximum revenue?

Answer 1

let x=the # of $1 decreases as well as the number of "2-people additions"

(400-1x)(300+2x)=0

120,000-300x+800x-2x²=0

120,000+500x-2x²=0

this is a parabola opening downward with its vertex being the maximum point

use -b/2a

-500/-4=125

125 $1 decreases and 125 "2-people increases will maximize revenue

(400-125)(300+250)

$275*550=$151,250 is the maximum revenue

124 yields $151,248 and 126 yields the same amount of revenue

the price charged is $275 each for 550 people