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anonymous
 5 years ago
An open box with a square bottom with a volume of 96 cubic inches is to be constructed. The bottom material costs three times as much per square inch as the side material. What dimensions will minimize the cost of the box?
anonymous
 5 years ago
An open box with a square bottom with a volume of 96 cubic inches is to be constructed. The bottom material costs three times as much per square inch as the side material. What dimensions will minimize the cost of the box?

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amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0volume of a box = length*width*height; and since its a square bottom, we could get away with: V = w^2 h = 96 the area of the sides of the box amount to 4 sides of w*h; and a bottom of w^2 4wh + w^2

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0lets find a good enough value for "h" to plug into this: We do that by using the volume 96 = w^2 h, h = 96/w^2

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.04w(96/w^2) + w^2(96/w^2) = A 384/w + 96 = A this is how much we have altogether in materials

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0the amount of material in the bottom cost 3times as much as the sides, so amount times price = cost..

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0an open box with the square bottom of having length 2 inches and the whole box having height of 24 inches

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0that is one possible scenario, yes :) but many dimensions can have the same area... i think

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i use differentiation to get the minimum cost

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0and my equation has an error in it, we dont multiply height to the second term 4w(96/w^2) + w^2 = A 384/w + w^2 = A yes, that is how you would optimize the problem :)

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0the bottom costs 3times as much....so lets assume that the price per side area is 1 and the price for the bottom would be 3 384/w + 3x^2 = total cost

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0derive and we get: 384/w^2 + 6w = C' 384 + 6w^3  = 0 w^2 6w^3 = 384 w^3 = 64 w = cbrt(64) w = 4 but i need to recheck my work ;)

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0it checks out in my books... the base is 4 by 4 16(h) = 96 h = 96/16 h = 6 4x4x6 would be the best option. lets compare that to 2x2x24 (2)(2)(3) + 4(2)(24) 12 + 192 = $204  (4)(4)(3) + 4(4)(6) 48 + 96 = $144

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0I would go with the 4x4x6 measurements :)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0haha yep u're right. I made a glaring mistake, argh

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0I was sweating it :) til I got to the end lol
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