1987 AMC 12 第 22 题

Question

湖面结冰时，一个球正漂浮在湖中。取出这个球时没有破坏冰面，留下的洞在冰面处宽

24

厘米、深

8

厘米。这个球的半径是多少厘米？

A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole

24

cm across at the top and

8

cm deep. What was the radius of the ball (in centimeters)?

8

12

13

8\sqrt3

6\sqrt6

取一个经过圆形洞中心的竖直截面

Take a vertical cross-section through the center of the circular hole

半弦长为

12

，而球心到冰面的距离为

r-8

The half-chord is

12,

while the center-to-ice distance is

r-8

在竖直截面中，宽

24

厘米的洞的一半是一条长为

12

的弦段。若球的半径为

r

，则球心到冰面的距离为

r-8

。由此形成的直角三角形满足

(r-8)^2+12^2=r^2\text{。}

解得

16r=208

，所以

r=13

。

因此，正确答案是 C。

In a vertical cross-section, half the

24

-cm hole is a chord segment of length

12.

If the sphere radius is

r,

the distance from its center to the ice plane is

r-8.

The resulting right triangle gives

(r-8)^2+12^2=r^2.

Solving yields

16r=208,

so

r=13.

Thus the correct answer is C.

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