http://www.neonlightssigns.info/sign-furniture-chair/
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Sign Furniture Chair
optimization and modeling problem 4 [calculus]?
you run a small furniture business. you sign a deal with a customer to deliver up to 400 chairs, the exact number to be determined by the customer later. the price will be $90 per chair up to 300 chairs, and above 300, the price for every additional chair will be reduced by $0.25 per chair (on the whole order) for every additional chair over 300 ordered. What are the largest and smallest revenues your company can make under this deal?
If I understand correctly, once the order exceeds
300 chairs, the price of *all* chairs is reduced.
For example, if the order were for 310 chairs, the
price for all chairs would be
Price per chair = $90 - (310-300)*.25
Price per chair = $90 - (10)*.25
Price per chair = $90 - 2.5
Price per chair = $87.5
So the revenue for the entire order would be:
Revenue = 310 * 87.5 = $27125
Working with that understanding, the equation for
price per chair for "x" number of chairs would be:
p = { for x<300: $90 }
p = { for x>=300: $90-(x-300)*.25 }
Revenue for "x" chairs would be computed as:
r = p * x
Since revenue linearly increases for increasing
numbers of chairs up to 300, so we're interested
in what happens *after* 300 chairs.
r = p * x
r = [90-(x-300)*.25] * x
r = [90-(0.25x-75)] * x
r = [90 - 0.25x + 75] * x
r = [165 - 0.25x] * x
r = 165x - 0.25x^2
To find the max/min, find the root of r'(x):
r'(x) = 165 - .5x
0 = 165 - .5x
.5x = 165
x = 330
r''(x) = -.5
Since r''(x) is negative, the function is concave
down, thus x=300 is a maximum point.
This can be verified with a spreadsheet, where the
revenue for 330 chairs would be:
r(330) = $27225
with smaller revenues for any larger or smaller "x".
To find the smallest revenues, we also need to
inspect the endpoints of the domain. It's not
clear to me what the minimum order is, but if it
is 0 chairs, then that is the minimum revenue ($0).
If the minimum is 300 chairs, the revenue is $27000.
The problem states that the maximum order is 400
chairs, which would generate a revenue of:
r(400) = 400 * $65 = $26000
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36"x96" FURNITURE SALE BANNER SIGN store signs sign sofa recliner chair dinette | |
|
US $61.74 | 4d 6h 11m |
|
72" RECLINER SALE BANNER SIGN furniture chairs sofa coffee tables lazyboy | |
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US $38.99 | 3d 3h 21m |
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RECLINER SALE BANNER SIGN furniture chairs sofa coffee tables lazyboy | |
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US $19.99 | 27d 8h 46m |
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FURNITURE SALE All Weather Banner Sign Chair Couch Bed | |
|
US $18.95 | 1d 7h 22m |
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48"x120" FURNITURE SALE BANNER SIGN store signs sign sofa recliner chair dinette | |
|
US $113.99 | 4d 6h 6m |
|
36"x96" RECLINER SALE BANNER SIGN furniture chairs sofa coffee tables | |
|
US $61.74 | 4d 4h 26m |
|
48"x120" RECLINER SALE BANNER SIGN furniture chairs sofa coffee tables | |
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US $113.99 | 4d 4h 21m |
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NEW FLASHING NEON SIGN "OFFICE FURNITURE" 14461 30"X17"X3" desk chair cabinets | |
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US $309.00 | 3d 22h 12m |
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ANTIQUES All Weather Banner Sign Furniture Chair China | |
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US $14.95 | 22h 20m |
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