1954 AMC 12 第 27 题

Question

一个正圆锥的底面圆与一个给定球的半径相同。圆锥体积是球体积的一半。圆锥的高与底面半径之比为：

A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:

\dfrac11

\dfrac12

\dfrac23

\dfrac21

\sqrt{\dfrac54}

答案：D

知识点：圆锥球体积

难度评级：1280

小提示：

使用公共半径

r

写出两个体积

Write both volumes using their common radius

r

大提示：

令

\frac13\pi r^2h

等于

\frac43\pi r^3

的一半

Set

\frac13\pi r^2h

equal to half of

\frac43\pi r^3

解答：

体积条件为

\frac13\pi r^2h=\frac12\left(\frac43\pi r^3\right)\text{。}

约去公因式可得

h=2r

，所以所求比为

2:1

。

因此，正确答案是 D。

The volume condition is

\frac13\pi r^2h=\frac12\left(\frac43\pi r^3\right).

Cancelling the common factors gives

h=2r,

so the requested ratio is

2:1.

Thus, the correct answer is D.

1954 AMC 12 第 27 题

答案：D

难度评级：1280

小提示：

大提示：

解答：

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