Solution to: Water Bucket
The surface of a cylinder with an open upper side is: O = pi×R2+2×pi×R×H, where R is the radius of the cylinder and H is the height. The volume is given, V = 30 = pi×R2×H liters (or dm3). The surface can then be written as O = pi×R2+2×V/R. If you take the first derivative of R and look where this derivative equals 0 you get Rmin = (V/pi)1/3. So the minimum surface is O = pi×Rmin2+2×pi×Rmin×H = 3×pi1/3×V2/3 or about 42.42 dm2.