A health club currently charges its 1500 clients monthly membership dues of $ 44. The board of directors decides to increase the monthly membership dues. Market research shows that each $ 1 increase in dues will result in the loss of 5 clients. How much should the club charge each month to optimize the revenue from monthly dues?

The club should charge $172 to optimize the revenue.

Step-by-step explanation:

Hi there!

The revenue can be calculated as the number of clients times the value of the membership dues. Let y be the revenue, then:

y = 1500 · 44

Let x be the increase in membership dues, then for a increse of x, the number of clients lost will be 5x:

y = (1500 - 5x) (44 + x)

y = 66000 - 220x + 1500x - 5x²

y = - 5x² + 1280x + 66000

It is a quadratic equation, a negative parabola. The maximum is the vertex of the parabola that is calculated as:

x = - b/2a

Where:

b = 1280

a = - 5

Then:

x = - 1280/2 · (-5)

x = 128

The club should charge 44 + 128 = $172 to optimize the revenue.