can anyone please help me set this problem up, I can do the math but I have tried for hours...

Question

Air fare club offers membership at $300.00, at least 50 people must join.For every member over 50 their fare will be reduced by $2 for every member. Due to space limits only 125 passengers will be allowed. how many memberships will maximize the revenue?

Answer 1

current rate = $300
number of members = 50

let the number of additonal member be n

so actual number of members = n+50
cost per member = 300 - 2n

Revenue = number of members x membership fee
= (n+50)(300 - 2n)
= 300n - 2n^2 + 15000- 100n
= -2n^2 + 200n + 15000
= -2(n^2 - 100n + 2500 - 2500 ) + 15000
= -2( (n-50)^2 - 2500) + 15000
= -2(n-50)^2 + 5000 + 15000
= -2(n-50)^2 + 20000

This is a parabola opening downward with a vertex of (50,20000)

So the maximum of Revenue is $20,000 when n = 50
So 50 addition to the original 50 should join.

The maximum revenue is obtained when 100 members are in.

check:
at 100 members cost is 300 - 2(50) = 200
revenue is 100(200) = 20000
at 99 members cost per member is 300-2(49) = 202
revenue is 99(202) = 19998 which is less than 20000
at 101 members cost per member is 300-2(51) = 198
revenue is 101(198) = 19998 , which is again less than 20000
My answer is correct.