Question 147149
A container is to be designed that is a rectangular box with an open top that will hold 108 cubic inches.  If x = the length of the side of the base, then express the surface area of the box as a function of X.  What is domain of this function and determine the dimensions of the box that minimizes the surface area of the box
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Assume the base is square with an area = x^2.
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Let h = height of the box
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The volume equation
x^2 * h = 108
h = {{{108/x^2}}}
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Surface area:
Bottom = x^2
4 sides = 4(x{{{108/x^2}}}) = {{{432/x}}}; area of the 4 sides
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S.A. = x^2 + {{{432/x}}}
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Graph: y = x^2 + {{{432/x}}}
{{{ graph( 300, 200, -6, 20, -30, 200, x^2+(432/x)) }}}
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Min surface area: x = 6
Find h = {{{108/6^2}}} = 3 inches
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min surface area of the box: 6^2 + 4(6*3) = 108 sq inches
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Interesting that the vol and surface area are equal.
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Domain of the equation: x > 0