1988 AMC 12 Problem 9

Question

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

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

Compare the room side with the table’s diagonal

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.

Problem 9 in other years