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A Barrel of Fun

The mean diameter of a barrel being equal to the head diameter + 2/3 the excess of the bung diameter over the head diameter (if staves are considerably curved), how many gallons does a barrel contain whose measurements are, head diameter 17 in., bung 20 inc., length 28 inc.?

Solution:

Source: Edward Olney's The Science of Arithmetic, 1876, p. 277.