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Question

A block of cheese in the shape of a rectangular solid measures 10 cm by 13 cm by 14 cm. Ten slices are cut from the cheese. Each slice has a width of 1 cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off?

Accepted Answer

Let the lengths of the three sides of the rectangular solid after the cutting be a,b,c, so that the desired volume is abc. Note that each cut reduces one of the dimensions by one, so that after ten cuts, a+b+c = 10 + 13 + 14 - 10 = 27. By the AM-GM inequality, a+b+c3 = 9 [3]abc abc 729. Equality is achieved when a=b=c=9, which is possible if we make one slice perpendicular to the 10 cm edge, four slices perpendicular to the 13 cm edge, and five slices perpendicular to the 14 cm edge.