Question 1058007
Let {{{ h }}} = the height of the box
Let {{{ w }}} = the width
{{{ 2w }}} = the length
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The cost of the base is:
{{{ 2w*w*13 }}}
{{{ 26w^2 }}}
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The area of the sides is:
{{{ 2*w*h + 2*( 2w )*h }}}
{{{ 2w*h + 4w*h = 6w*h }}}
The cost of the materials for sides is:
{{{ 6w*h*7 = 42w*h }}}
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Also given:
{{{ w*2w*h = 12 }}}
{{{ 2w^2h = 12 }}}
{{{ h = 12/( 2w^2 ) }}}
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Let {{{ C }}} = cost of materials
The cost for materials is:
{{{ 26w^2 + 42w*h }}}
{{{ 26w^2 + 42w*( 12/( 2w^2 ) ) }}}
{{{ C = 26w^2 + 252/w }}}
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I don't know if this is for a calculus course, but
that's the only way I can solve it.
Take the derivative and set  = zero
{{{ 52w - 252/w^2 = 0 }}}
{{{ 52w = 252/w^2 }}}
{{{ 52w^3 = 252 }}}
{{{ w^3 = 4.8462 }}}
{{{ w = 1.6923 }}}
and
{{{ C[ min ] = 26w^2 + 252/w }}}
{{{ C[ min ] = 26*1.6923^2 + 252/1.6923 }}}
{{{ C[ min ] = 74.4573 + 148.9098 }}}
{{{ C[min] = 223.367 }}}
The minimum cost is $223.37
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Check:
Is the 2nd derivative positive? ( that makes it a min )
{{{ 52 - 252*(-2)/w^3 }}}
{{{ 52 + 504/w^3 }}} yes, it's positive
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Here's the plot of {{{ C(w) }}}:
{{{ graph( 400, 400, -2, 10, -400, 400, 26x^2 + 252/x ) }}} 
I may have the right answer. Get another opinion on this