health club currently charges its 1900 clients monthly membership dues of $ 42. 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 $211 to optimize the revenue.

Step-by-step explanation:

Hi there!

The current revenue of the health club is calculated by multiplying the number of clients by the membership due value.

If y = revenue, then:

y = 1900 · 42

So, if the memberiship dues increase by $1, 5 clients will be lost. Then, if the dues increase by a value of x, then the number of lost clients will be 5x. If such an increase is applied, the revenue will be:

y = (1900 - 5x) (42 + x)

Apply distributive property:

y = 79800 + 1900x - 210x - 5x²

y = - 5x² + 1690x + 79800

The function is a negative parabola, then, its maximum will be the vertex of the parabola that can be calculated using the following formula:

x = - b/2a

Where:

b = 1690

a = - 5

Then:

x = - 1690 / (2 · - 5)

x = 169

The club should charge 169 + 42 = $211 to optimize the revenue.