2003 AMC 10B 第 17 题

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

\frac{\pi r^2 h}{3}

，半径为

r

的球体积为

\frac{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

\frac{\pi r^2 h}{3},

and a sphere with radius

r

has volume

\frac{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

小提示：

圆锥和球有相同的半径

r

。

The cone and sphere share the same radius

r

大提示：

令

\tfrac34 \cdot \tfrac43 \pi r^3

等于圆锥体积

\tfrac13 \pi r^2 h

。

Set

\tfrac34 \cdot \tfrac43 \pi r^3

equal to the cone volume

\tfrac13 \pi r^2 h

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\text{。}

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

小提示：

大提示：

解答：

其他年份的第 17 题