Question

Fitness The gym charges $180 for a yearly membership. There are currently1000 members. For every $5 increase in price, the gym will lose 10 members.How much should the gym charge to maximize its revenue?

Answer 1

Let the number of $5 increases be m. We get the following two equations:

new charge = 180 + 5n

membership = 1000 - 10n

thus, the revenue would be:

revenue = (180 + 5n)( 1000 - 10n)

Applying the distributive property, this is equivalent to:

`180000+3200-50n^2`

If we factor this equation, we obtain:

`-50\mleft(n^2-64n-3600\mright)`

notice that this is a parabola and the vertex will yield our answer. The n of the vertex is:

`(64)/(2)=32`

Applying this data into the first two equations, we get:

new charge = 180 + 5n = 180 + 5(32)= 340

membership = 1000 - 10n = 1000 - 10(32) = 680

So that, we can conclude that the correct answer is:

The new fee is $340.