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 need to find the slope and the y-intercept.

We can use the given information to find the slope:

For Sig Ep fraternity, they sold 240 shirts at $5 per shirt, so their data point is (240, 5).
For Ep Sig fraternity, they sold 480 shirts at $2 per shirt, so their data point is (480, 2).

Using the formula for slope (m), we can calculate it as:
m = (y2 - y1) / (x2 - x1)
m = (2 - 5) / (480 - 240)
m = -3 / 240
m = -1/80

Now, let's find the y-intercept. We can use either data point, and we'll use Sig Ep fraternity's data point (240, 5).

Using the point-slope form of a linear equation (y - y1) = m(x - x1), we can substitute the values:
(y - 5) = (-1/80) (x - 240)

Expanding and simplifying this equation, we get the linear demand equation:
y = (-1/80)x + 6

So, the linear demand equation for Yoda vs. Alien T-shirts is y = (-1/80)x + 6.

(b) To calculate the weekly profit (P), we need to subtract the total cost (C) from the total revenue (R). The fraternities are charged a weekly fee of $500 for the use of the Student Center, so this will be part of the cost.

Let's say the unit price is x dollars.

The total revenue is given by:
R = x * quantity

The quantity can be determined by substituting the unit price into the demand equation:
y = (-1/80)x + 6

To calculate the quantity, we substitute:
quantity = (-1/80)x + 6

So, the total revenue can be expressed as:
R = x * [(-1/80)x + 6]

The total cost (C) consists of two components:
- The cost of producing each T-shirt, which we'll denote as C1 (per T-shirt cost).
- The weekly fee charged by the university, which is $500.

We are not given the cost of producing each T-shirt, so let's represent it as C1 dollars.

The cost per week (C) can be expressed as:
C = C1 * quantity + $500

Now, we can calculate the weekly profit (P) as:
P = R - C

Let's simplify this equation by substituting the expressions we derived above:
P = (x * [(-1/80)x + 6]) - (C1 * [(-1/80)x + 6] + $500)

To find the unit price that maximizes the weekly profit, we need to take the derivative of P with respect to x, set it equal to zero, and solve for x. However, we need more information about the cost per T-shirt (C1) to proceed further.