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 R_min = (V/pi)^1/3. So the minimum surface is O = pi×R_min²+2×pi×R_min×H = 3×pi^1/3×V^2/3 or about 42.42 dm².

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