marginal cost
asked Nov 9, 2016 in Calculus Answers by maybin

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Profit = revenue - cost.

x items are sold for xP=200x-0.01x^2.

The cost of producing x items is C=50x+20000.

Profit=200x-0.01x^2-50x-20000=150x-0.01x^2-20000.

To maximise the profit we differentiate this: 150-0.02x and equate to zero, so x=150/0.02=50*150=7500.

To solve without using calculus:

150x-0.01x^2-20000=-(0.01x^2-150x+20000)=

-0.01(x^2-15000+2000000)=

-0.01(x^2-15000+7500^2+2000000-7500^2)=

-0.01((x-7500)^2+2000000-56250000)=-0.01((x-7500)^2-54250000)=

0.01((54250000-(x-7500)^2).

This expression has the greatest value when x=7500.

 

answered Nov 9, 2016 by Rod Top Rated User (442,460 points)
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