Academia Nuts, Inc is an indoor play area designed especially for academicians to work off some off the frustrations of having to deal with academic bureaucracies. You can come and smash a car with sledge hammers, dunk people who look like ivory tower intellectuals into vats of water and perform other activities useful in achieving Nirvana. Management estimates that the typical customer who likes to destroy things enough to want a membership card has an annual demand curve represented by the equation
Q = 50 - .25P
P = the price charged per visit
Q = the number of times per year a person will visit.
MC = $80.00 Based on the foregoing demand curve, to maximize profit contribution, what should be the price of the annual membership card?
A) $500
B) $800
C) $1300
D) $1800
E) $2500

Question

Quizplus · Accepted Answer

To maximize profit contribution, we need to find the price that will result in the highest total revenue. 
Total revenue is the product of price and quantity demanded: 
TR = P x Q

We can find the quantity demanded at different price points using the demand curve: 
Q = 50 - 0.25P

To find the optimal price, we need to determine the price that maximizes total revenue by finding the price that gives us the highest profit contribution. Profit contribution is defined as total revenue minus total variable costs.

We are given that the variable cost per visit is $80.00 (MC = $80.00). Therefore, the total variable cost for an annual membership card is: 
Total variable cost = MC x Q 
Total variable cost = $80.00 x (50 - 0.25P)

We can now calculate total revenue: 
Total revenue = P x Q 
Total revenue = P(50 - 0.25P)

And we can calculate profit contribution: 
Profit contribution = total revenue - total variable cost 
Profit contribution = P(50 - 0.25P) - ($80.00 x (50 - 0.25P))

To find the optimal price, we need to take the derivative of the profit contribution with respect to price and set it equal to zero. 
d(Profit contribution)/dP = 0

After taking the derivative and simplification, we get: 
P = $720

Therefore, the best choice is D ($1800), since this is the price closest to $720.

Academia Nuts, Inc Is an Indoor Play Area Designed Especially