A small fitness center that offers only personal training services has the follo

Question

A small fitness center that offers only personal training services has the following demand and cost parameters:

Demand: The fitness center has found that it has some discretion in pricing—that is, it can raise price marginally without drastic reductions in volume. Based on statistical estimates of demand and assuming that external factors stay constant (e.g., price of competitors' services, income levels, etc.), the following relationship exists between the hourly rate for a personal training session (P) and the number of sessions demanded per day (Q):

P = 140 – Q.

Costs: The fitness center finds that its variable costs (e.g., labor) increase at a constant rate of $40 with each additional training session provided per day. Fixed costs such as rent are equal to $200 per day. This yields the following total variable cost (TVC) and total fixed cost (TFC) equations:

TVC = 40Q.
TFC = 200.

Explanation / Answer

To just break even, you would need to produce 3 training sessions.

From the profit = total revenue - total cost column,

the highest possible profit is $2300.00 which can be achievedby charging $90.00 per session and selling 50 sessions.

This managerial decision maximizesprofit in column,

L.Profit = Total Revenue – Total Cost

2300 = P*Q - TC(Q)

2300 = (140-Q)*Q - (200+40Q)

2300 = 140Q - Q^2- 200 + 40Q

Q^2 = 180Q - 200 - 2300

Q^2  = 180Q - 2100

Q^2 - 180Q = - 2100

- Q . (Q - 180) = - 2100

-Q^2 = - 2280

Q =  2280 = 47.7

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