Question 581762
The Perimeter is:
{{{ P = 2W + 2L }}}
{{{ 1000 = 2W + 2L }}}
{{{ 500 = W + L }}}
{{{ L = 500 - W }}}
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The Area is:
{{{ A = W*L }}}
{{{ A = W*( 500 - W ) }}}
{{{ A = -W^2 + 500W }}}
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First of all, when the coefficient of the
squared term is negative, then there is a maximum
and not a minimum.
When the equation is in the form
{{{ y = ax^2 + ba + c }}}, the max or min is
always at {{{ x = -b/(2a) }}}, or, in this case,
{{{ W[max] = -500/(2*(-1)) }}}
{{{ W[max] = 250 }}}
and I know that
{{{ L = 500 - W }}}
{{{ L[max] = 500 - 250 }}}
{{{ L[max] = 250 }}}
The largest area that the farmer can enclose
is {{{ 250*250 }}} = 62,500 ft2
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I can prove this by changing the dimensions
slightly, but still keeping the same Perimeter
{{{ 249*251 }}} = 62,499 ft2
{{{ 248*252 }}} = 62,496 ft2 ( even less )