Question 969172
{{{V=pi*r^2*h=410}}}
{{{SA=2pi*r^2+2pi*r*h}}}
From the volume,
{{{h=410/(pi*r^2)}}}
Substitute into the surface area equation,
{{{SA=2pi*r^2+2pi*r*(410/(pi*r^2))}}}
{{{SA=2pi*r^2+820/r}}}
Now surface area is only a function of the radius.
Take the derivative and set it equal to zero to find a min.
{{{d(SA)/dr=4pi*r-820/r^2=0}}}
{{{4pi*r=820/r^2}}}
{{{r^3=820/(4pi)}}}
{{{r^3=205/pi}}}
{{{r=(205/pi)^(1/3)}}}
{{{r=4.03}}}{{{cm}}}
Then,
{{{h=410/(pi*4.03^2)}}}
{{{h=8.04}}}{{{cm}}}