Maximizing and Minimizing Revenue

shelwa

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Nov 11, 2011
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2
The problem states:

"You are a small furniture business owner. you sign a deal with a customer to deliver up to 400 chairs, the exact number TBD by the customer later. The price will be $90 per chair up to 300 chairs, and above 300, the price will be reduced by .25 per chair (on the whole order) for each additional chair over 300 ordered. What are the largest and smallest revenues your company can make under this deal?"

I thought I could get the maximum revenue, but for every chair above 300 the order gets reduced by $0.25 per chair. I'm not sure how to tackle this, any suggestions?
 
First step ALWAYS, LABEL your variables.

Let C = number of chairs, and
R = revenue.

Second step ALWAYS, create one or more equations or inequations that RELATE your variables

R = 90C if C
4_de44c582df9d8d29dbbd70aca311c641.png
300.

R = [90 - .25(C - 300)] * C if C
1_cedf8da05466bb54708268b3c694a78f.png
300.

Proceed, but note that the function describing R is NOT differentiable at C = 300.

Can you explain how you got the second part of the piecewise function, please? Also, I'm a bit confused about how to get the minimum. since on the left there's a smaller minimum than on the right.
 
Let x=the number of additional chairs.

Then, the cost per extra chair would be \(\displaystyle (90-.25x)\)

The number ordered would then be \(\displaystyle 300+x\)

Revenue \(\displaystyle (90-.25x)(300+x)\)

This is what is to be maximized.

Per the constraint that there will be at most 400 chairs ordered, then \(\displaystyle x\leq 100\)

If you optimize, you will find that x will be within the desired range.
 
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