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 480 per week at $2 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.What is the weekly profit P as a function of the unit price x?
Determine how much the fraternities should charge to obtain the largest possible weekly profit.
To construct a linear demand equation for Yoda vs. Alien T-shirts, we need to find the relation between the quantity sold and the unit price. We have two data points: Sig Ep sold 240 shirts at $5 per shirt, and Ep Sig sold 480 shirts at $2 per shirt.
(a) Let's use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
Using the first data point (240 shirts at $5 per shirt), we have (x1, y1) as (5, 240). Plugging this into the equation, our equation becomes: y - 240 = m(x - 5).
Using the second data point (480 shirts at $2 per shirt), we have (x2, y2) as (2, 480). Plugging this into the equation and solving for m:
480 - 240 = m(2 - 5)
240 = -3m
m = -80
So our linear demand equation is: y = -80x + b, where b is the y-intercept. To find b, we can plug in one of the data points (e.g., x = 5, y = 240):
240 = -80(5) + b
240 = -400 + b
b = 640
Therefore, our linear demand equation is: y = -80x + 640. This equation represents the quantity sold per week (y) as a function of the unit price (x).
To obtain the weekly revenue R as a function of the unit price x, we multiply the quantity sold (y) by the unit price (x):
R(x) = x * y = x * (-80x + 640) = -80x^2 + 640x
(b) The weekly profit P is the revenue R minus the weekly fee the fraternities have to pay to the administration. In this case, the fee is $500.
P(x) = R(x) - 500 = -80x^2 + 640x - 500
To find the unit price that maximizes the weekly profit, we need to find the vertex of the parabolic profit function P(x).
The x-coordinate of the vertex of a quadratic function in the form ax^2 + bx + c is given by x = -b / (2a).
For P(x) = -80x^2 + 640x - 500, a = -80 and b = 640.
x = -640 / (2 * -80) = -640 / -160 = 4
So the unit price that maximizes the weekly profit is $4 per shirt.
To calculate the corresponding profit, substitute x = 4 into the profit function:
P(4) = -80(4)^2 + 640(4) - 500 = -1280 + 2560 - 500 = 780
Therefore, the fraternities should charge $4 per shirt to obtain the largest possible weekly profit of $780.