Question

一张 $8'\times10'$ 的桌子位于一间正方形房间的角落，如下面图 $1$ 所示。主人希望把桌子移到图 $2$ 所示的位置。房间边长为 $S$ 英尺。若移动时不能倾斜或拆卸桌子，能完成所需移动的最小整数 $S$ 是多少？

An $8'\times10'$ table sits in the corner of a square room, as in Figure $1$ below. The owners desire to move the table to the position shown in Figure $2.$ The side of the room is $S$ feet. What is the smallest integer value of $S$ for which the table can be moved as desired without tilting it or taking it apart?

$11$

$12$

$13$

$14$

$15$

答案：C

知识点：rectangle diagonal rotation geometric bounds

难度评级：1540

小提示：

在转过四分之一圈的过程中，一条对角线会与一对相对的墙垂直

During a quarter-turn, a diagonal becomes perpendicular to a pair of opposite walls

大提示：

将房间边长与桌子的对角线比较

Compare the room side with the table’s diagonal

解答：

桌子的对角线长为 $\sqrt{8^2+10^2}=\sqrt{164}$ ，它在 $12$ 与 $13$ 之间。转动过程中，这条对角线必须能放进房间，所以 $S\ge\sqrt{164}$ 。反过来，矩形可以在以其对角线为直径的圆内转动，因此这个界也足够。最小整数 $S$ 为 $13$ 。

所以正确答案是 C。

The table diagonal is $\sqrt{8^2+10^2}=\sqrt{164},$ which is between $12$ and $13.$ During the turn this diagonal must fit across the room, so $S\ge\sqrt{164}.$ Conversely, a rectangle can rotate inside a circle whose diameter is its diagonal, so that bound is sufficient. The least integer $S$ is $13.$

Thus the correct answer is C.

1988 AMC 12 第 9 题

答案：C

难度评级：1540

小提示：

大提示：

解答：

其他年份的第 9 题