Problema Solution
Brooke opens a gym and charges the members $300 per year to join. At that price, she gets 1000 members. For each $2 increase in price she can expect to lose 6 members. For each $2 increase in price she can expect to lose 6 members. How much should she charge to maximize its revenue. What is the gym'maximum revenue.
Answer provided by our tutors
price: $300; number of members: 1000;
revenue y = 300*1000
price: $300; number of members: 1000;
for 2x dollars increase in the price she losses 6x members
revenue y = (300 + 2x)(1000 - 6x) simplifying we have
y = -12x^2 + 200x + 300000
is a parabolic function with negative quotient in front of x^2 that is a = -12 <0 thus the parabolic function has maximum in its vertex
see the graph of the function here
y max = c - (b^2/4a) where a = -12, b = 200, c = 300000
y max = 300000 - 200^2/(4(-12))
y max = $300,833.33 is the gym's maximum revenue.
we find the the point for which the function has extreme using
y' = 0
- 24x + 200 = 0
x = 200/24
to maximize the revenue she should charge
300 + (200/24)*2 = $316.666