Question 1057295
The cylinder has two basic measurements.  Height and diameter  (or "length" and radius).  


v for volume,
{{{v=h*pi*r^2}}}


s for surface area,
{{{s=2*pi*r^2+h*pi*2*r}}}


You are given the volume, in cubic feet,
{{{pi*h*r^2=50}}}
{{{h=50/(pi*r^2)}}}
Substitute for h in the surface area formula.


{{{s=2pi*r^2+(50/(pi*r^2))2*pi*r}}}


{{{s=2pi*r^2+100/r}}}


Minimizing the area s means finding {{{ds/dr}}} and setting equal to 0.  
{{{4pi*r+100(-1)r^-2=0}}}
{{{4pi*r-100/r^2=0}}}
{{{4pi*r^3/r^2-100/r^2=0}}}
{{{(4pi*r^3-100)/r^2=0}}}
From here, you are interested in the NUMERATOR being 0.
{{{4pi*r^3-100=0}}}
{{{pi*r^3-25=0}}}
{{{pi*r^3=25}}}
{{{r^3=25/pi}}}
{{{highlight(r=root(3,25/pi))}}}, and from this you can evaluate h.