AMC10 2003 B

AMC10 2003 B · Q17

AMC10 2003 B · Q17. It mainly tests Fractions, 3D geometry (volume).

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 $4\pi r^3 /3$.)

一个冰淇淋甜筒由一个香草冰淇淋球和一个直径与球相同的圆锥组成。如果冰淇淋融化，它正好充满圆锥。假设融化的冰淇淋体积占冷冻冰淇淋体积的75%。圆锥的高度与其半径的比是多少？（注：半径为$r$、高度为$h$的圆锥体积为$\pi r^2 h /3$，半径为$r$的球体积为$4\pi r^3 /3$。）

(A) 2 : 1 2 : 1

(B) 3 : 1 3 : 1

(D) 16 : 3 16 : 3

(E) 6 : 1 6 : 1

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) Let $r$ be the radius of the sphere and cone, and let $h$ be the height of the cone. Then the conditions of the problem imply that \[ \frac{3}{4}\left(\frac{4}{3}\pi r^{3}\right)=\frac{1}{3}\pi r^{2}h,\ \text{so }h=3r. \] Therefore, the ratio of $h$ to $r$ is $3:1$.

（B）设 $r$ 为球和圆锥的半径，设 $h$ 为圆锥的高。则题目的条件意味着 \[ \frac{3}{4}\left(\frac{4}{3}\pi r^{3}\right)=\frac{1}{3}\pi r^{2}h,\ \text{因此 }h=3r. \] 因此，$h$ 与 $r$ 的比为 $3:1$。

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