Manufacturer of luxury chairs determines that in order to sell x chairs, the pri

Question

Manufacturer of luxury chairs determines that in order to sell x chairs, the price per chair must be p(x) = 1000 - x.(So, if they want to sell 300 chairs, they should set the price to be 1000-300 = $700.) The manufacturer also knows that the total cost of producing x chairs is given by C(x) = 3000 + 20x. Therefore, the profit to the manufacturer if they produce and sell x chairs will be: F(x) = (revenue) - (cost) = x(1000 - x) - (3000 + 20x) How many chairs should they produce and sell to make the largest possible profit? What should they set the price to be? What will their profit be, at that price?

Explanation / Answer

a.

We need to maximise the profit function. So we differentiate with respect to x.

F(x)=x(1000-x)-(3000+20x)=-x^2+980x-3000

F'(x)=-2x+980

F'(x)=0 gives, -2x+980=0,x=490

F''(x)=-2<0

Hence, x=490 is point of maxima

SO ,x=490 is the number of chairs to be sold to maximise profit

b.

Price per chair would be,p(490)=1000-490=510 per chair

c.

Profit will be

F(490)=-490^2+980*490-3000=237100

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