Question

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 240 shirts in a week at $5 per shirt, but Ep Sig treasurer Julius has solid research indicating that it is possible to sell 400 per week at $3 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?

Answer 1

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.

• y = 640 - 80x ⇒ demand equation
• xy = - 80x² + 640x ⇒ weekly revenue

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?

• $780

Explanation:

first, we must determine the slope = (400 - 240) / (3 - 5) = 160 / -2 = -80

the demand equation:

y - 240 = -80 (x - 5)

y = -80x + 400 + 240

y = 640 - 80x

total weekly revenue:

xy = -80x² + 640x

xy - 500 = -80x² + 640x - 500

max. profit ⇒ x = -640 / (2 x -80) = -640 / -160 = 4

maximum weekly profit = -80($4²) + 640($4) - $500 = -$1,280 + $2,560 - $500 = $780