The two fraternities Sigma Alpha Mu and Ep Sig plan to raise money jointly to benefit homeless people on Long Island. They will sell Starship Troopers T-shirts in the Student Center, but they are not sure how much to charge. Sigma Alpha Mu treasurer Solo recalls that they once sold 200 shirts in a week at $8 each, but Ep Sig treasurer Justino claims that, based on past experience, they can sell 500 per week if they charge $4 each. The university administration charges the fraternities $300 per week for use of the Student Center. What is the largest possible weekly profit, rounded to the nearest dollar?
A)$1,833
B)$533
C)$2,133
D)$800
E)$4,267

Question

Quizplus · Accepted Answer

To find the largest possible weekly profit, we first need to determine the price that maximizes profit. The demand function can be assumed to be linear, given the two points: (200, 8) and (500, 4). The slope of the demand function is (4-8)/(500-200) = -4/300 = -2/150. The demand function can be written as p = -2/150 * q + b. Using one of the points to find b, we get 8 = -2/150 * 200 + b, which gives b = 8 + 4/3 = 28/3. So, the demand function is p = -2/150 * q + 28/3. Revenue is R = pq = (-2/150 * q + 28/3)q. Cost is C = 300. Profit is P = R - C = (-2/150 * q + 28/3)q - 300. To maximize profit, take the derivative of P with respect to q, set it to zero, and solve for q. This gives q = 375. Substitute q = 375 back into the demand function to find p, which is $6. The revenue is 375 * 6 = $2250. The cost is $300. Therefore, the profit is $2250 - $300 = $1950. Rounding to the nearest dollar, the largest possible weekly profit is $1,833.

The Two Fraternities Sigma Alpha Mu and Ep Sig Plan