Question 1128389
The proportion of the sector to a full circle determines the area:
{{{ A = (theta/360)(pi)r^2 }}}   <br>

The value of {{{ theta }}} is the ratio of the arc length to the circumference of a full circle.  Noting that the arc length is  {{{ l-2r }}}:
{{{ theta / 360 =  (l-2r) / (2(pi)r) }}}<br>

Substituting the latter into the former:
{{{ A = ((l-2r)/2(pi)r) (( pi) r^2 ) }}}
{{{ A = (lr/2) - r^2 }}} <br>

Taking the derivative of A with respect to r:
{{{ dA/dr = l/2 - 2r }}} <br>

Setting this to zero:
{{{ l/2 - 2r = 0 }}}  —>  {{{ highlight( r = l/4 ) }}}   (<<<—— that's an "ell" on top)   <br>

——
Check:
{{{ d^2A/dr^2 = -2 }}} —>  the function is concave down, thus the point dA/dr=0 we found is a maximum.