AMC10 2009 A

AMC10 2009 A · Q11

AMC10 2009 A · Q11. It mainly tests Linear equations, 3D geometry (volume).

One dimension of a cube is increased by 1, another is decreased by 1, and the third is left unchanged. The volume of the new rectangular solid is 5 less than that of the cube. What was the volume of the cube?

一个立方体的边长有一个维度增加1，另一个维度减少1，第三个维度保持不变。新矩形体的体积比立方体的小5。立方体的体积是多少？

(A) 8 8

(B) 27 27

(D) 125 125

(E) 216 216

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Let \(x\) be the side length of the cube. Then the volume of the cube was \(x^3\), and the volume of the new solid is \(x(x + 1)(x - 1) = x^3 - x\). Therefore \(x^3 - x = x^3 - 5\), from which \(x = 5\), and the volume of the cube was \(5^3 = 125\).

答案（D）：设 \(x\) 为立方体的棱长，则立方体的体积为 \(x^3\)，新立体的体积为 \(x(x+1)(x-1)=x^3-x\)。因此 \(x^3-x=x^3-5\)，解得 \(x=5\)，所以立方体的体积为 \(5^3=125\)。

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