Two fraternities, Sig Ep and Ep Sig, plan to raise money jointly to benefit home

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 160 shirts in a week at $6 per shirt, but Ep Sig treasurer Julius has solid research indicating that it is possible to sell 240 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.

R(x) =

(b) The university administration charges the fraternities a weekly fee of $300 for use of the Student Center. Find the weekly profit P as a function of the unit price x.

P(x) =

Determine how much the fraternities should charge to obtain the largest possible weekly profit.
x = $  per T-shirt

What is the largest possible weekly profit?
$

Explanation / Answer

Solution:

put price on the x axis, and quantity on the y axis

you now have to points
(6,160) for 160 at $6
(4,240) for 240 at $4

find the slope that conects them
(6,160) - (4,240) = {2, -80}--->m = -40 pretty steep, but consistant with the elasticity of t-shirts

sub in an orderd pair
240 = (-40)4+b
240 = -160+b
400 = b

(a)
q(x) = -40x + 400
x is the price, q is the quantity
xq = revenue
R(x) = x(-40x+400)
R(x) = -40x^2 + 400x

(b)
profit = revenue - costs
P = R - 300
P(x) = (-40x^2 + 400x) - 300
P(x) = -40x^2 + 400x - 300

max profit will be the max point on your graph of P. you can pull the quantity off of there.
x=5
so you charge $5.00/shirt

P(5) = -40(5)^2 + 400*5 - 300 = 700
the max profit is $700.00

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