Question 880814
First find the volume of the cylinder.
{{{V=pi*R^2*L=150}}}
Then first the surface area of the cylinder.
{{{SA=2*pi*R^2+2*pi*R*L}}}
Now substitute from the volume so that the surface area is now only a function of one variable.
{{{L=150/(piR^2)}}}
So then,
{{{SA=2*pi*R^2+2*pi*R*(150/(piR^2))}}}
{{{SA=2*pi*R^2+300/R}}}
To minimize the surface area, take the derivative with respect to R and set it equal to zero.
{{{d(SA)/dR=4*pi*R-300/R^2}}}
{{{4*pi*R-300/R^2=0
{{{4*pi*R=300/R^2
{{{R^3=300/(4*pi)}}}
{{{R=(175/(2*pi))^(1/3)}}}
So then you can work backwards to find L.