a producer can sell x items per week at a price P=200-0.01x kwacha and it cost C=50X+20000 to make x items. what is the most profitable number to make?

Question

marginal cost

Answer 1

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.