The curved surface area of a cone is slant height * perimiter of base/2,

so if the height is h, and the radius of the base is r, we have slant height:

s = sqrt(h^2+r^2)

So the area of canvas required is:

A = sqrt(h^2+r^2) (2 pi r)/2

The volume is:

V = (1/3) (pi r^2) h = 1000,

so:

h = 3000/(pi r^2),

and area of canvas:

A = sqrt([3000/(pi r^2)]^2+r^2) (2 pi r)/2

But if A is minimise then so is A^2, so you may as well minimise:

A^2 = ([3000/(pi r^2)]^2+r^2) pi^2 r^2

RonL