Question 1151231
Make 2 lengths add up to {{{ 1 }}} and
be in golden ratio to each other.
{{{ x / ( 1 - x ) = ( 1 - x ) / 1 }}}
{{{ x = ( 1 - x )^2 }}}
{{{ x = 1 - 2x + x^2 }}}
{{{ x^2 - 3x + 1 = 0 }}}
{{{ x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
{{{ x = ( -(-3) +- sqrt( 9 - 4*1*1 )) / ( 2*1) }}}
{{{ x = ( 3 - sqrt( 5 )) / 2 }}}
and
{{{ 1 - x = 2/2 - ( 3 - sqrt(5) ) / 2 }}}
{{{ 1 - x = ( 2 - 3 + sqrt(5) ) / 2 }}}
{{{ 1 - x = ( -1 + sqrt(5) ) / 2 }}}
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Let the length of the shorter side = {{{ y }}}
{{{ 2y - 5 }}} = length of the longer side
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{{{ y / ( 2y - 5 ) = (  ( 3 - sqrt( 5 )) / 2 ) / (  ( -1 + sqrt(5) ) / 2 ) }}}
{{{ y / ( 2y - 5 ) = (  ( 3 - sqrt( 5 ) ) ) / (  ( -1 + sqrt(5) ) ) }}}
{{{ y / ( 2y - 5 ) = ( 3 - 2.2361 ) / ( -1 + 2.2361 ) }}}
{{{ y / ( 2y - 5 ) = .7639 / 1.2361 }}}
{{{ y / ( 2y - 5 ) = .618 }}}
{{{ y = .618*( 2y - 5 ) }}}
{{{ y = 1.236y - 3.09 }}}
{{{ .236y = 3.09 }}}
{{{ y = 13.093 }}}
and
{{{ 2y - 5 = 2*13.093 - 5 }}}
{{{ 2y - 5 = 26.186 - 5 }}}
{{{ 2y - 5 = 21.186 }}}
To the nearest foot, the dimensions are
13 x 21
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check the math & get 2nd opinion if needed