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2009 AMC 12A Problem 5

Problem 5 of 25EasierAlgebraGeometry

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 dimensions x+1,x + 1, x−1,x - 1, and x,x, so its volume is 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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