Question

一个长方体的总表面积为 $94$ 平方英寸。它所有棱长之和为 $48$ 英寸。它所有体对角线长度之和是多少英寸？

A rectangular box has a total surface area of $94$ square inches. The sum of the lengths of all its edges is $48$ inches. What is the sum of the lengths in inches of all of its interior diagonals?

$8\sqrt{3}$

$10\sqrt{2}$

$16\sqrt{3}$

$20\sqrt{2}$

$40\sqrt{2}$

答案：D

知识点：长方体表面积代数变形

难度评级：1840

解答：

设三条边长为 $x, y, z$ 则 $xy+yz+zx = 47$ ，且 $x+y+z = 12$ 。因此 $\begin{gathered} x^2+y^2+z^2 = (x+y+z)^2 \\ {}- 2(xy+yz+zx) \\ = 144 - 94 \\ = 50. \end{gathered}$

$4$ 条体对角线中的每一条长度为 $\sqrt{x^2+y^2+z^2} = \sqrt{50} = 5\sqrt2$ ，所以总长度为 $4 \cdot 5\sqrt2 = 20\sqrt2$ 。

所以正确答案是 D。

Let the edges be $x, y, z.$ Then $xy+yz+zx = 47$ and $x+y+z = 12.$ Therefore $\begin{gathered} x^2+y^2+z^2 = (x+y+z)^2 \\ {}- 2(xy+yz+zx) \\ = 144 - 94 \\ = 50. \end{gathered}$

Each of the $4$ interior diagonals has length $\sqrt{x^2+y^2+z^2} = \sqrt{50} = 5\sqrt2,$ so their total length is $4 \cdot 5\sqrt2 = 20\sqrt2.$

Thus, the correct answer is D.

2014 AMC 12B 第 14 题

答案：D

难度评级：1840

解答：

其他年份的第 14 题