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 140 shirts in a week at $8 per shirt, but Ep Sig treasurer Julius has solid research indicating that it is possible to sell 420 per week at $4 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 $300 for use of the student center. Find the weekly profit Pas a function of the unit price x.

Answer 1

Answer:

a)

Equation of demand

q = -280p + 2380

where p=price, q=quantity sold

Equation of revenue (quantity sold times the unitary price p*q)

R(p) = p(-280p + 2380)

b)

Equation of profit (Revenue - fee)

P(p) = p(-280p + 2380) - 300

Step-by-step explanation:

a)

Let us consider the price p as the independent variable and the quantity q of T-shirts sold at the price p as the variable dependent on the price.

To construct a linear demand equation, we construct the equation of the line that passes for the points

(p1, q1) = (4, 420)

(p2, q2) = (8, 140)

This line has slope

(140-420)/(8-4) = -280

So, the equation in the slope-intercept form would be

q = -280p + b

To compute b, we replace any of the points on the equation, for example (8, 140)

140 = -280*8 + b ===> b = 2380

and the quantity-demand line would be

q = -280p + 2380

The weekly revenue would be given by

R(p) = pq = p(-280p + 2380)

where p is the price of the T-shirts.

b)

The weekly profit would be given by

P = R(p) - 300 = p(-280p+2380) -300