A commodity has a demand function modeled by p = 103 − 0.5x and a total...

Question

Question

A commodity has a demand function modeled by p = 103 − 0.5x and a total cost function modeled by C = 30x + 31.75, where x is the number of units.

(a) What price yields a maximum profit?

(b) When the profit is maximized, what is the average cost per unit? (Round your answer to two decimal places.)

Math Precalculus

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Answer 1

Answer #1

p(x) = 103−0.5x; p is the demand function and x is the number of units

c(x) = 30x+31.75; c is the cost function

Revenue is given by R(x)=price*number of units sold =p*x=(103−0.5x)(x)=103x-0.5x²

Profit is q(x)=revenue−cost= (103x-0.5x²)-(30x+31.75)=-0.5x²+73x-31.75

for maximum profit, q'(x)=0 and q''(x)<0

q(x)=-0.5x²+73x-31.75

q'(x)=-0.5(2x)+73-0

q'(x)=-x+73

q''(x)=-1<0 i.e. When q'(x)=0, the profit is maximum

q'(x)=0

or, -x+73=0

or, x=73

Price=p(73)=103-0.5(73)=66.5

Price of 66.5 yields maximum profit.

Average Cost = Total Cost / x

or, Average cost=30x+31.75/x

or, Average cost=(30*73+31.75)/73

or,Average cost= 30.43