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：

其他年份的第 27 题