1991 AMC 12 第 8 题

Question

液体

X

不与水混合。在没有阻碍时，它会在水面上铺展成厚度为

0.1

厘米的圆形薄膜。一个长、宽、高分别为

6

厘米、

3

厘米和

12

厘米的长方体容器装满液体

X

。把其中的液体倒在一大片水面上，所得圆形薄膜的半径是多少厘米？

Liquid

X

does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film

0.1

cm thick. A rectangular box measuring

6

cm by

3

cm by

12

cm is filled with liquid

X.

Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film?

\frac{\sqrt{216}}{\pi}

\sqrt{\frac{216}{\pi}}

\sqrt{\frac{2160}{\pi}}

\frac{216}{\pi}

\frac{2160}{\pi}

令容器中液体的体积等于圆形薄膜的体积

Equate the volume in the box to the volume of the thin circular film

薄膜的体积为

\pi r^2(0.1)

The film volume is

\pi r^2(0.1)

液体的体积为

6\cdot3\cdot12=216

立方厘米。因此

0.1\pi r^2=216\text{，}

所以

r^2=\frac{2160}{\pi}

，并且

r=\sqrt{\frac{2160}{\pi}}

。

因此正确答案为 C。

The liquid volume is

6\cdot3\cdot12=216

cubic centimeters. Thus

0.1\pi r^2=216,

so

r^2=\frac{2160}{\pi}

and

r=\sqrt{\frac{2160}{\pi}}.

Thus the correct answer is C.

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