CASE 2 The owner of a small local health club has estimated the demand for healt
ID: 2617463 • Letter: C
Question
CASE 2 The owner of a small local health club has estimated the demand for health club memberships, q, to be q0.06p + 46 where q is the number of members in the club and p is the annual membership fee she currently charges. Annual operating costs amount to only $5000 per year, and each member costs the club $100 per year for medical evaluation and personal training. What is the profit-maximizing number of members, and what range (the minimum and the maximum] of memberships make sense to remain in business?Explanation / Answer
P = 766.67 - 16.67 Q
Profit = Revenue - costs
Profit = ( 766.67 - 16.67 Q)Q - [ $ 5000 + $ 100Q]
taking the first derivative of above function
dp/dq = 766.67 - 33.34Q - 100
taking second derivative of above function
d2p/dq2 = -33.34
equating first derivative to zero
766.67 - 33.34Q - 100 = 0
Q = 20
The profit maximizing number of members = 20
___________________________________
(766.67 - 16.67 Q)Q - [ $ 5000 + $ 100Q] = profit
Equate the profit equating to zero
766.67Q - 16.67Q2 - 5000 - 100Q = 0
-16.67Q2 + 666.67Q - 5000 = 0
16.67Q2 - 666.67 + 5000 = 0
Q = 666.67 +- sqrt 666.672 - 4 * 16.67 * 5000 / 2 * 16.67
Q = 666.67 +- 333.24 / 2*16.67
Q = 10 units 0r Q = 30 units
The range of members that makes sense to remain in business is 10 to 30 members.
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