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.
(a) To construct a linear demand equation, we can use the slope-intercept form y = mx + b, where y represents the quantity sold per week, x represents the unit price, m represents the slope, and b represents the y-intercept.
From the given information, Sig Ep sold 240 shirts per week at $5 per shirt, and Ep Sig sold 480 shirts per week at $2 per shirt.
We can calculate the slope using the formula: m = (y2 - y1) / (x2 - x1).
Using Sig Ep's data as (x1, y1) = (5, 240) and Ep Sig's data as (x2, y2) = (2, 480), we find:
m = (480 - 240) / (2 - 5) = 240 / -3 = -80.
Substituting these values into the slope-intercept form, we have y = -80x + b.
To find the y-intercept (b), we can use one of the data points. Using Sig Ep's data, when x = 5 and y = 240:
240 = -80 * 5 + b,
240 = -400 + b,
b = 640.
Therefore, the linear demand equation for Yoda vs. Alien T-shirts is:
y = -80x + 640,
where y represents the quantity sold per week, and x represents the unit price.
(b) The weekly revenue R is calculated as R = xy, where x represents the unit price, and y represents the quantity sold per week.
Since y = -80x + 640, we can substitute this equation into the revenue formula:
R = x(-80x + 640),
R = -80x^2 + 640x.
To obtain the weekly profit P, we deduct the university administration fee of $500 from the revenue:
P = R - 500,
P = -80x^2 + 640x - 500.
To determine the unit price that maximizes the weekly profit, we can find the vertex of the quadratic function. The x-coordinate of the vertex can be calculated using the formula: x = -b / (2a).
For our equation P = -80x^2 + 640x - 500, we have a = -80 and b = 640.
x = -640 / (2 * -80) = -640 / -160 = 4.
Therefore, the fraternities should charge $4 per shirt to obtain the largest possible weekly profit.