Question 1150427
Let {{{ w }}} = the width of the rectangle
{{{ 4w }}} = the total of the {{{ w }}} lengths of fence
{{{ ( 600 - 4w ) / 2 }}} = the length of the rectangle
Let {{{ A }}} = the total area of the rectangle
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{{{ A = 4w*( ( 600 - 4w ) / 2 ) }}}
{{{ A = 2w*( 600 - 4w ) }}}
{{{ A = -8w^2 + 1200w}}}
This is a parabola with a maximum peak
{{{ w[pk] = -1200/(2*(-8)) }}}
{{{ w[pk] = 75 }}}
Plug this result back into equation
{{{ A = -8w^2 + 1200w}}}
{{{ A[max] = -8*75^2 + 1200*75}}} ( formula for vertex of parabola )
{{{ A[max] = -45000 + 90000 }}}
{{{ A[max] = 45000 }}}
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{{{ 45000/w[pk] = 45000/75 }}}
{{{ 45000/w[pk] = 600 }}}
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The dimensions that maximize area are:
75' x 600'
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check:
{{{ graph( 400, 400, -20, 200, -5000, 50000, -8x^2 + 1200x ) }}}