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posted by  Novaworks on 11/2/2009 8:59:54 PM  |  status: Closed  |  Earned Karma: 1145

A box with a square bas

Course Textbook Chapter Problem Needs by
Calculus N/A N/A N/A 11/2/2009 at 1:00:00 PM
Question Details:
A box with a square base and open top must have a volume of 4000 cm3. Find the dimensions of the box that minimize the amount of material used.
1? cm (length and width)
2? cm (height)
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posted by Altair on 11/2/2009 9:13:52 PM  |  status: Live
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Response Details:
let the side of the square be x, the heigh be y
we have: V = x2 y  = 4000cm2   =>y = 4000cm2 / x2
the surface area is S = x2 + 4xy  = x2 + 4*4000 cm3 /x
to find th mininum S, we have:
S' = 2x - 16000 cm2 /x2
let S' = 0, => 2x - 16000 cm2 /x2 = 0
=>x3 = 8000 cm3
=>x = 20 cm
y =  4000cm2 / x2 = 10 cm
answer: length and width: 20 cm
heigh : 10cm

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posted by zsm28 on 11/2/2009 9:14:14 PM  |  status: Live
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Response Details:
side of base = x, height = h
x2h = V = 4000, so h = V/x2
total area A(x) = x2 + 4xh = x2 + 4x*V/x2 = x2 + 4V/x
A'(x) = 2x - 4V/x2 = 0
x = (2V)1/3 = 20 cm
h = 10 cm
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