Let the radius of the cylinder is r and height is h.

Volume of cylinder `= 512m^3`

`pi*r^2*h = 512`

` h = 512/(pi*r^2)`

The amount of material needed can be represent by the area of the material.

Surface area of cylinder `(A) = pi*r^2+2*pi*r*h`

`A = pi*r^2+2*pi*r`

`A = pi*r^2+2*pi*r*512/(pi*r^2)`

`A = pi*r^2+1024/r`

When the material amount is least or maximum then `(dA)/(dr) = 0`

`(dA)/dr = 2pi*r+1024*(-1/r^2)`

When `(dA)/(dr) = 0;`

`2pi*r+1024*(-1/r^2) = 0`

`r^3 = 512/pi`

`r = 5.462`

When A is a minimum then `(d^2A)/(dA^2) >0` at` r = 5.462`

`(d^2A)/(dA^2)`

`= 2pi+1024*2/r^3`

since r>0; `(d^2A)/(dA^2)>0 ` always. So we have a minimum for A at `r = 5.462`

*Radius of cylinder for minimum material = 5.462*

**Height of cylinder for minimum material = `512/(pi*5.462^2)` = 5.462**

*So the height and radius of the cylinder is equal when the area of the material is minimum.*

Note:

It is assumed that the area of material for joints and welds are negligible or constant in all cases.