Question

一个冰淇淋甜筒由一个香草冰淇淋球和一个与球直径相同的直圆锥组成。若冰淇淋融化，它正好装满圆锥。假设融化后的冰淇淋体积是冷冻时体积的 $75\%$ 。圆锥的高与半径之比是多少？

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $75\%$ of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius?

$2 : 1$

$3 : 1$

$4 : 1$

$16 : 3$

$6 : 1$

答案：B

知识点：圆锥球体积

难度评级：1490

小提示：

球和圆锥有相同的半径 $r$ ；球体积为 $\dfrac{4}{3}\pi r^3$

The sphere and cone share the same radius $r;$ sphere volume is $\dfrac{4}{3}\pi r^3$

大提示：

令球体积的 $75\%$ 等于圆锥体积 $\dfrac{1}{3}\pi r^2 h$

Set $75\%$ of the sphere’s volume equal to the cone’s volume $\dfrac{1}{3}\pi r^2 h$

解答：

设共同半径为 $r$ ，圆锥高为 $h$ 。融化后的冰淇淋装满圆锥，所以 $\frac{3}{4}\cdot\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 h\text{。}$

化简得 $\pi r^3 = \dfrac{1}{3}\pi r^2 h$ ，所以 $h = 3r$ ，比为 $3 : 1$ 。

因此，正确答案是 B。

Let $r$ be the common radius and $h$ the cone’s height. The melted ice cream fills the cone, so $\frac{3}{4}\cdot\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 h.$

This simplifies to $\pi r^3 = \dfrac{1}{3}\pi r^2 h,$ so $h = 3r,$ a ratio of $3 : 1.$

Thus, the correct answer is B.

2003 AMC 12B 第 13 题

答案：B

难度评级：1490

小提示：

大提示：

解答：

其他年份的第 13 题