anonymous
  • anonymous
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?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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amistre64
  • amistre64
volume 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
  • amistre64
lets 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
  • amistre64
4w(96/w^2) + w^2(96/w^2) = A 384/w + 96 = A this is how much we have altogether in materials

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amistre64
  • amistre64
the amount of material in the bottom cost 3times as much as the sides, so amount times price = cost..
anonymous
  • anonymous
an open box with the square bottom of having length 2 inches and the whole box having height of 24 inches
amistre64
  • amistre64
that is one possible scenario, yes :) but many dimensions can have the same area... i think
anonymous
  • anonymous
i use differentiation to get the minimum cost
amistre64
  • amistre64
and 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
  • amistre64
the 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
  • amistre64
derive 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
  • amistre64
it 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
  • amistre64
I would go with the 4x4x6 measurements :)
anonymous
  • anonymous
haha yep u're right. I made a glaring mistake, argh
amistre64
  • amistre64
I was sweating it :) til I got to the end lol

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