How much should the gym charge to maximize its revenue? What is the maximum revenue?

Question

If a gym charges its members $300 per year to join, they get 1000 members. For each $2 increase in price they can expect to lose 5 members.

Answer 1

For each increase of $2, the gym expects to lose 5 people.

 

Right now, the revenue is 300 * 1000 = 300,000.

 

the new dollar amount can be represented as 300+2x. (x is the number of increases and decreases)

the new population is represented by 1000 - 5x since it loses 5 people for every increase.

 

So R(x) = (300+2x)(1000-5x)

using foil you get

 

R(x) = 300,000 -1500x + 2000x -10x² which equals 300,000 + 500x - 10x² .

 

This is a parabola opening downward ( since -10 is the coefficient of the x² term) which means it has a maximum value at its turning point.

 

The turning point is on the axis of symmetry ( x=-b/2a) so the axis of symmetry for this parabola is

x=-500/-20 which is x=25.

 

R(25) = (350)(875) = 306,250

 

So the coordinates of this maximum in (25, 306250).

 

So if there are 25 raises making the price $350, even though the customers will go down, the dollar intake will go up to 306,250.

 

Hope this helped.