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2009 AMC 10A Problem 11

Problem 11 of 25IntermediateAlgebraGeometry

One dimension of a cube is increased by 1,1, another is decreased by 1,1, and the third is left unchanged. The volume of the new rectangular solid is 55 less than that of the cube. What was the volume of the cube?

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Solution

Let the cube have side length x.x. The new solid has volume x(x+1)(x−1)=x3−x.x(x+1)(x-1) = x^3 - x. Setting this equal to x3−5x^3 - 5 gives x3−x=x3−5,x^3 - x = x^3 - 5, so x=5.x = 5. The cube’s volume is 53=125.5^3 = 125. Thus, the correct answer is D.

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Concepts: difference of squares · volume · linear equation

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.