-10x+100x+6000

Question

A hall charges $30 per person for a sports banquet when 200 people attend. For every 10 extra people that attend, the hall will decrease the price by $1 per person. What number of people will maximize the revenue for the hall?

Answer 1

let x represent the number of groups of 10 people

now you are making $30*200=$6000

(200+10x)(30-1x)=0

6000+300x-200x-10x^2=0

6000+100x-10x^2=0

600+10x-x^2=0

this is a parabola that opens downward so there is a maximum point, (h,k), where h=-b/2a which gives the maximum revenue, k

-b/2a=-10/-2=5

5 groups of 10 people, or 50 people, will maximize the revenue

check: 200+50=250 people and $30-$5=$25 per person

250*$25=$6250

if you have 4 groups of 10 people...

240*$26=$6240, which is less money

if you have 6 groups of 10 people...

260*$24=$6240, again less money