Question

A health club currently charges its 1700 clients monthly membership dues of $ 45. 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 7 clients. How much should the club charge each month to optimize the revenue from monthly​ dues?

Answer 1

$135

Step-by-step explanation:

Given:

Total clients = 1700

Membership dues = $45

Increase in monthly dues = $1

Loss of clients per dollar increase = 7 clients

Thus,

let x be the number of dollar increases

therefore,

clients lost will be 7x

so the revenue function will be

f(x) = charges × Number of clients

or

f(x) = ( 45 + x ) × ( 1700 - 7x )

or

f(x) = 90000 - 315x + 1700x - 7x²

or

f(x) = 90000 + 1385x - 7x²

now,

for point of maxima or minima

differentiating with respect to x, we get

f'(x) = 0 + 1385 - 14x = 0

or

14x = 1385

or

x = 98.92 ≈ 98

thus,

to optimize the revenue from monthly dues the club should charge

( $45 + $90 ) = $135