Question 875784
For a cylinder,
{{{V[0]=pi*R^2*H}}}
For the surface area of the tank (amount of metal used to make it),
{{{A=2*pi*R^2+2*pi*R*H}}}
From above,
{{{H=V[0]/(pi*R^2)}}}
Substituting,
{{{A=2*pi*R^2+2*pi*R*(V[0]/(pi*R^2))}}}
{{{A=2*pi*R^2+(2V[0])/R}}}
To find the minimum amount of metal take the derivative of A and set it to zero,
{{{dA/dR=4*pi*R-(2V[0])/R^2=0}}}
{{{4*pi*R=(2V[0])/R^2}}}
{{{V[0]=2pi*R^3}}}
From above,
{{{pi*R^2*H=2pi*R^3}}}
{{{H=2R}}}