Question 1147993
Let {{{ h }}} = the height
{{{ h*x*( x + 4 ) = 100 }}}
{{{ h = 100/( x^2 + 4x ) }}}
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The surface area for top and bottom is:
{{{ S[1] = 2*x*( x+4 ) }}}
The area for the front and the back is:
{{{ S[2] = 2*( x+4 )*h }}}
The area for the sides is:
{{{ S[3] = 2x*h }}}
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{{{ S[1] + S[2] + S[3] = 2x^2 + 8x + ( 2x + 8 )*h + 2x*h }}}
{{{ S[1] + S[2] + S[3] = 2x^2 + 8x + h*( 2x + 8 + 2x ) }}}
{{{ S[1] + S[2] + S[3] = 2x^2 + 8x + ( 100/( x^2 + 4x ))*( 4x + 8 ) }}}
{{{ S[1] + S[2] + S[3] = 2x^2 + 8x + ( 400x + 800 ) / ( x^2 + 4x ) }}}
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One way to check this is to pick a value for {{{ x }}}
{{{ x = 9 }}}
{{{ x + 4 = 13 }}}
{{{ h = 100/( x^2 + 4x ) }}}
{{{ h = 100/( 9^2 + 4*9 ) }}}
{{{ h = 100/117 }}}
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{{{ S[1] = 2*x*( x+4 ) }}}
{{{ S[1] = 2*9*( 9+4 ) }}}
{{{ S[1] = 234 }}}
and
{{{ S[2] = 2*( x+4 )*h }}}
{{{ S[2] = 2*( 9+4 )*(100/117) }}}
{{{ S[2] = 2600/117 }}}
and
{{{ S[3] = 2x*h }}}
{{{ S[3] = 18*(100/117) }}}
{{{ S[3] = 1800/117 }}}
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{{{ 234 + 4400/117 =  2x^2 + 8x + ( 400x + 800 ) / ( x^2 + 4x ) }}}
{{{ 234 + 4400/117 =  2*9^2 + 8*9 + ( 400*9 + 800 ) / ( 9^2 + 4*9 ) }}}
{{{ 234 + 4400/117 =  2*9^2 + 8*9 + ( 3600 + 800 ) / ( 81 + 36 ) }}}
{{{ 234 + 4400/117 =  162 + 72 + 4400 / 117 }}}
{{{ 234 + 4400/117 =  234 + 4400 / 117 }}}
OK
check the math