## Solution to: Water Bucket

The surface of a cylinder with an open upper side is:
O = pi×R^{2}+2×pi×R×H,
where R is the radius of the cylinder and H is the height. The volume is given,
V = 30 = pi×R^{2}×H liters (or dm^{3}).
The surface can then be written as O = pi×R^{2}+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_{min}^{2}+2×pi×R_{min}×H = 3×pi^{1/3}×V^{2/3} or about 42.42 dm^{2}.

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