T-Shirt Profit Two fraternities, Sig Ep and Ep Sig, plan to raise money jointly to benefit homeless people on Long Island. They will sell Yoda vs. Alien T-shirts in the student center, but are not sure how much to charge. Sig Ep treasurer Augustus recalls that they once sold 400 shirts in a week at $8 per shirt, but Ep Sig treasurer Julius has solid research indicating that it is possible to sell 600 per week at $4 per shirt.
a. Based on this information, construct a linear demand equation for Yoda vs. Alien T-shirts, and hence obtain the weekly revenue R as a function of the unit price x.
b. The university administration charges the fraternities a weekly fee of $500 for use of the Student Center. Write down the monthly profit P as a function of the unit price x, and hence determine how much the fraternities should charge to obtain the largest possible weekly profit. What is the largest possible weekly profit? HINT [See Example 4.]
Example 4:
As the operator of YSport Fitness gym, you calculate your demand equation to be
q = −0.06p + 84
where q is the number of members in the club and p is the annual membership fee you charge.
a. Your annual operating costs are a fixed cost of $20,000 per year plus a variable cost of $20 per member. Find the annual revenue and profit as functions of the membership price p.
b. At what price should you set the annual membership fee to obtain the maximum revenue? What is the maximum possible revenue?
c. At what price should you set the annual membership fee to obtain the maximum profit? What is the maximum possible profit? What is the corresponding revenue?
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