Solution to: Water Bucket

The surface of a cylinder with an open upper side is: O = π × R² + 2 × π × R × H, where R is the radius of the cylinder and H is the height. The volume is given, V = 30 = π × R² × H liters (or dm³). The surface can then be written as O = π × R² + 2 × V/R. If you take the first derivative of R and look where this derivative equals 0 you get R_min = (V/π)^1/3. So the minimum surface is O = π × R_min² + 2 × π × R_min × H = 3 × π^1/3 × V^2/3 or about 42.42 dm².

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