the gym charges 180$ for a yearly membership. there are currently 1000 members. for every 5$ increase the gym will lose 10 members. how much should the gym charge to maximize its revenue?

Answer 1

current charge -- $180
membership ---- 1000

Let the number of $5 increases be m
new charge = 180 + 5n
membership = 1000 - 10n

revenue = (180+5n)(1000-10n)
= 180000 + 3200 - 50n^2
= -50(n^2 - 64n - 3600)
this is a parabola and the vertex will yield our answer
the n of the vertex is 64/2 = 32

so the new charge should be 180 + 5(32) = 340
for a membership of 1000 - 10(32) = 680

check:
current revenue = 180(1000) = 180,000
at "best", revenue = 340(680) = 231,000

my answer is reasonable

Answer 2

456