Question 1029100
You are asked to minimize expense
Let {{{ s }}} = the length of one of the sides
that face north and south and costs $1 / ft
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Let {{{ A }}} = area enclosed by fences
{{{ A = s*( 5000/s ) }}}
So, the sides are {{{ s }}} and {{{ 5000/s }}}
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Let {{{ P }}} = the perimeter of the area fenced in
The perimeter of the area is:
{{{ P = 2s + 2*( 5000/s ) }}}
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Let {{{ C }}} = the cost of fencing in the area
{{{ C = 1*( 2s ) + 2*2*( 5000/s ) }}}
{{{ C = 2s + 20000/s }}}
{{{ C1 = 2 - 20000 / s^2 }}}
Set the 1st derivative, {{{ C1 = 0 }}}
{{{ 0 = 2 - 20000 / s^2 }}}
{{{ 20000 / s^2 = 2 }}}
{{{ s^2 = 10000  }}}
{{{ s = 100 }}}
and
{{{ 5000/s = 5000/100 }}}
{{{ 5000/s = 50 }}}
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Plug these results back into original equation
{{{ C = 2s + 20000/s }}}
{{{ C = 2*100 + 20000/100 }}}
{{{ C = 200 + 200 }}}
{{{ C = 400 }}}
The least expensive fence costs $400
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check answer:
{{{ A = s*( 5000/s ) }}}
{{{ A = 100*50 }}}
{{{ A = 5000 }}}
OK -here's the plot of Cost, {{{ C }}} and {{{ s }}}
{{{ graph( 500, 500, -30, 200, -100, 600, 2x + 20000/x ) }}}