Question 1197132
<pre>
Total area of open cyclinder is:
A(r) = {{{ pi*r^2 + 2*pi*r*L }}}   where r is radius and L is length of side

The volume is:
V(r) = {{{ pi*r^2*L }}}

We can relate L to r by the given information:
  {{{ pi*r^2 = 2*pi*r*L = 462 }}} ===>  {{{ L = (462 - pi*r^2) / (2*pi*r) }}}

So 
{{{ V(r) =  pi*r^2 * (462-pi*r^2)/(2*pi*r) }}}

which reduces to:
{{{ V(r) = 231*r - (pi/2)*r^3 }}}

Now the derivative of V with respect to r is:
{{{  dV/dr = 231 - ((3*pi)/2)*r^2 }}}

Setting dV/dr to 0 and solving for r,  {{{ highlight( r = 7\.0 ) }}} cm 

Check by plugging in some values near 7.0cm for r, and compute corresponding volume:
r = 6.8  ==>  L = 7.4  ==>  V = 1074.9cm^3
r = 7.0  ==>  L = 7.0  ==>  V = 1077.6cm^3   <<< max capacity
r = 7.2  ==>  L = 6.6  ==>  V = 1076.5cm^3