Question

一个立方体的一条边增加 $1$ ，另一条边减少 $1$ ，第三条边不变。新的长方体体积比原立方体体积少 $5$ 。原立方体的体积是多少？

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?

$8$

$27$

$64$

$125$

$216$

答案：D

知识点：平方差体积一次方程

难度评级：1100

解答：

设立方体边长为 $x$ 。新长方体的体积为 $x(x+1)(x-1) = x^3 - x.$

新长方体的体积为 $x^3 - 5$ ，所以 $x^3 - x = x^3 - 5$ ，得到 $x = 5$ 。

原立方体体积为 $5^3 = 125$ 。

所以正确答案是 D。

Let the cube have side length $x.$ The new solid has volume $x(x+1)(x-1) = x^3 - x.$

Setting this equal to $x^3 - 5$ gives $x^3 - x = x^3 - 5,$ so $x = 5.$

The cube's volume is $5^3 = 125.$

Thus, the correct answer is D.

2009 AMC 10A 第 11 题

答案：D

难度评级：1100

解答：

其他年份的第 11 题