Question

You run a small furniture business. You sign a deal with a customer to deliver up to 500 chairs, the exact number to be determined by the customer later. The price will be $120 per chair up to 400 chairs, and above 400 the price will be reduced by $0.25 per chair (on the whole order) for every additional chair over 400 ordered. What is the largest revenue your company can make under this deal?

I've found the minimum to be zero for this problem because they could sell no chairs at all and from what I can deduct from the problem the maximum they can make should be selling 400 chairs at $120 each which would be $48000 but my online turn in keeps telling me that's wrong and I don't understand why that is, please help.

Answer 1

let the number of chairs over 400 be x

now:
number of chairs = 400
price per chair = 120
Revenue = 48000

after production increase:
number of chairs = 400+x
price per chair = 120 - .25x

Revenue
= R
= (400+x)(120-.25x)
= 48000 + 20x - .25x^2
which is a downwards opening parabola
the x of the vertex is -20/-.5 = 40

so if x = 40
R = 440(120-.25(40)) = $48,400