The latest demand equation for your \"Banjos Rock\" T-shirts is given by q = 40x

Question

The latest demand equation for your "Banjos Rock" T-shirts is given by q = 40x + 3600 where q is the number of shirts you can sell in one week if you charge x dollars per shirt. When you charge x dollars per shirt, your weekly cost function (in dollars) is given by C(x) = 900x + 106312.5

a) Find the weekly profit as a function of the price per shirt x. p(x) = -40x^2+4500x-106312.5 (solved)

(b) Determine the unit price you should charge to break even. Enter the smaller value first. You break even, when you charge ___ or ___ dollars per shirt. (please show work for part b)

Explanation / Answer

a) We have given q(x)=-40x+3600 where q is the number of shirts and C(x)=900x + 106312.5

we know the revenue is the number of shirts sold q(x) and their price x

R(x)=x*q(x)=x(-40x+3600)=-40x^2+3600x

the profit is difference between revenue and cost function

P(x)=R(x)-C(x)

=-40x^2+3600x-(900x + 106312.5)

=-40x^2+4500x- 106312.5

P(x)=-40x^2+4500x-106312.5

b) we know that break-even point is when revenue=cost

P(x)=revenue-cost=-40x^2+4500x-106312.5=0

-40x^2+4500x-106312.5=0

by comparing with ax^2+bx+c=0 we get x=[-b+-sqrt(b^2-4ac)]/2a

we get a=-40,b=4500,c=-106312.5

x=[-4500+-sqrt((4500)^2-4*(-40)*(-106312.5))]/(2*(-40))

=[-4500+-sqrt(20250000-17010000)]/(-80)

=[-4500+-1800]/(-80)

x=[-4500+1800]/(-80) or x=[-4500-1800]/(-80)

x=33.75 or x=78.75

you break even, when you charge 33.75 or 78.75 dollars per shirt.

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