A health club offers memberships at the rate of $300, provided that at least 50 people join. For each member in excess of 50, the membership fee will be reduced by $2 for all members. Due to space limitations, at most 125 memberships will be sold. How many memberships should the club sell in order to maximize its revenue?

Question

Quizplus · Accepted Answer

Let x be the number of members above 50. Then, the membership fee is:
$300 - $2x, for 50 + x members.
The revenue generated is the product of the membership fee and the number of members:
Revenue = (300 − 2x)(50 + x) = 15000 − 100x + 2x^2
We want to find the value of x that maximizes the revenue. To do so, we take the derivative of the revenue function with respect to x and set it equal to zero:
d(Revenue)/dx = −100 + 4x = 0
4x = 100
x = 25
Therefore, 50 + x = 75 is the number of members that should be sold to maximize revenue. 
Note that if x > 25, then the total number of members will exceed 125, which is not allowed. Therefore, the answer is B) 100.

A Health Club Offers Memberships at the Rate of $300