Question

一个冰淇淋蛋筒由一个香草冰淇淋球和一个直圆锥组成，圆锥直径与球的直径相同。若冰淇淋融化后正好装满圆锥，并假设融化后冰淇淋体积是冷冻时体积的 $75\%$ ，则圆锥高与半径之比是多少？（注：半径为 $r$ 、高为 $h$ 的圆锥体积为 $\pi r^2 h / 3$ ，半径为 $r$ 的球体积为 $4\pi r^3 / 3$ 。）

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? (Note: A cone with radius $r$ and height $h$ has volume $\pi r^2 h / 3,$ and a sphere with radius $r$ has volume $4\pi r^3 / 3.$ )

$2 : 1$

$3 : 1$

$4 : 1$

$16 : 3$

$6 : 1$

答案：B

知识点：体积球圆锥

难度评级：1370

解答：

融化后的体积等于圆锥体积，所以 $\dfrac34 \cdot \dfrac43 \pi r^3 = \dfrac13 \pi r^2 h.$

化简得 $\pi r^3 = \dfrac13 \pi r^2 h$ ，所以 $h=3r$ ，高与半径之比为 $3:1$ 。

所以正确答案是 B。

The melted volume equals the cone's volume, so $\dfrac34 \cdot \dfrac43 \pi r^3 = \dfrac13 \pi r^2 h.$

Simplifying gives $\pi r^3 = \dfrac13 \pi r^2 h,$ so $h=3r.$ The ratio of height to radius is $3:1.$

Thus, the correct answer is B.

2003 AMC 10B 第 17 题

答案：B

难度评级：1370

解答：

其他年份的第 17 题