Question 309443
The volume of an open box with a square base and rectangular sides us 250 cubed inches.
 if sides are double thickness and the bottom is triple thickness, what size box will use the least amount of material?
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Surface area of an open box with a square base
x = side of the square base
h = height of the box
S.A = x^2 + 4(x*h)
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Surface area required with a triple thickness bottom and double thickness sides
S.A. = 3x^2 + 8(x*h)
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Vol = x^2*h
x^2*h = 250
h = {{{250/x^2}}}
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S.A = 3x^2 + 8(x*h)
Substitute for h
S.A. = 3x^2 + 8(x*{{{250/x^2}}})
S.A. = 3x^2 + 8({{{250/x}}})
S.A. = 3x^2 + {{{2000/x}}}
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Plot this equation
{{{ graph( 300, 200, -6, 16, -200, 1000, 3x^2 + (2000/x) ) }}} 
You can see min material for surface area occurs when x = 7
Find the height
h = {{{250/7^2}}}
h = 5.1 inches
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Least amt of material used in a 7 by 7 by 5.1 inch box
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