AMC12 2003 B

AMC12 2003 B · Q13

AMC12 2003 B · Q13. 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?

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。如果冰淇淋融化，它将正好填满圆锥。假设融化后的冰淇淋体积是冻结时体积的 $75\%$。圆锥的高与其半径之比是多少？

(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

Let $r$ be the common radius of the sphere and the cone, and $h$ be the cone’s height. Then \[75\% \cdot \left(\frac 43 \pi r^3\right) = \frac 13 \pi r^2 h \Longrightarrow h = 3r\] Thus $h:r = 3:1$ and the answer is $\boxed{B}$.

设球与圆锥的公共半径为 $r$，圆锥的高为 $h$。则 \[75\% \cdot \left(\frac 43 \pi r^3\right) = \frac 13 \pi r^2 h \Longrightarrow h = 3r\] 因此 $h:r = 3:1$，答案为 $\boxed{B}$。

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