Question 505917
The largest area for a given perimeter is always a square,
but you can prove it
let the vertical side = {{{y}}}
Let the horizontal side = {{{x}}}
If the perimeter is {{{P}}},
and the area is {{{A}}}
{{{ P = 2x + 2y }}}
{{{ P = 100 }}} cm
{{{ 100 = 2x + 2y }}}
(1) {{{ x + y = 50 }}}
{{{ A = x*y }}}
{{{ y = 50 - x }}}
{{{ A = x*(50 - x) }}}
{{{ A = -x^2 + 50x }}}
Since the coefficient of the squared term is negative
this curve has a maximum, not a minimum.
The vertex is at {{{ x = -b/(2a) }}}
{{{ b = 50 }}}
{{{ a = -1 }}}
{{{ x = -50/(2*(-1)) }}}
{{{ x = 25 }}}
and, since
{{{ y = 50 - x }}}
{{{ y = 25 }}}
So the sides are all 25 cm and the max area is a square