1988 AMC 12 第 9 题

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

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

将房间边长与桌子的对角线比较

Compare the room side with the table’s diagonal

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.

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