Question 1075632
Let each of the 2 base dimensions = {{{ x }}}
Let the height dimension = {{{ y }}}
The area of the 4 sides = {{{ 4*x*y }}}
The area of the base = {{{ x^2 }}}
The area of the top is also {{{ x^2 }}}
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{{{ 2*( x^2 + 4*x*y ) + 4*x^2 = 72 }}}
{{{ 2x^2 + 8*x*y + 4x^2 = 72 }}}
{{{ 8*x*y  = 72 - 6x^2 }}}
{{{ y = 9/x - (3/4)*x }}}
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The volume is:
{{{ V = x^2*( 9/x - (3/4)*x ) }}}
{{{ V = 9x - (3/4)*x^3 }}}
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If volume {{{ V }}} is maximized, then the cost of
construction will still be $72 total.
I will assume that this is a calculus problem, and
I will set the slope of {{{ V }}} equal to zero.
slope = {{{ 9 - (9/4)*x^2 = 0 }}}
{{{ (9/4)*x^2 = 9 }}}
{{{ x^2 = 4 }}}
{{{ x = 2 }}}
and
{{{ y = 9/x - (3/4)*x }}}
{{{ y = 9/2 - (3/4)*2 }}}
{{{ y = 4.5 - 1.5 }}}
{{{ y = 3 }}}
The dimensions in feet are:
2,2,3
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check answer:
Here's the plot of volume, {{{ V }}}, on vertical axis
and base dimension, {{{ x }}}, on the horizontal axis.
{{{ graph( 400, 400, -5, 5, -20, 20, 9x - (3/4)*x^3  ) }}} 
It looks like {{{ V }}} is maximized when {{{ x = 2 }}}
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Get a 2nd opinion if needed