2008 AMC 12B 第 11 题

Question

一座圆锥形山的底部在海底，高为

8000

英尺。山体体积的顶端

\tfrac{1}{8}

在水面以上。山底处的海水深度是多少英尺？

A cone-shaped mountain has its base on the ocean floor and has a height of

8000

feet. The top

\tfrac{1}{8}

of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet?

4000

2000(4 - \sqrt{2})

6000

6400

7000

水面以上的部分是一个与整座山相似的小圆锥。

The above-water part is a smaller cone similar to the whole mountain

体积比

\tfrac{1}{8}

是高度比的立方；深度等于总高度减去水上圆锥的高度。

The volume ratio

\tfrac{1}{8}

is the cube of the height ratio; the depth is the total height minus the above-water cone’s height

水面以上的部分是一个与整座山相似的圆锥，其体积为总体积的

\tfrac{1}{8}

。由于体积按长度的立方缩放，水上圆锥的高度是全高的

\sqrt[3]{\tfrac{1}{8}} = \tfrac{1}{2}

。

所以水面以上的高度为

8000 \cdot \tfrac{1}{2} = 4000

英尺。

山底处的海水深度就是水下高度，等于

8000 - 4000 = 4000

英尺。

因此，正确答案是 A。

The part above the water is a cone similar to the whole mountain, with volume

\tfrac{1}{8}

of the total. Since volume scales as the cube of length, the above-water cone’s height is

\sqrt[3]{\tfrac{1}{8}} = \tfrac{1}{2}

of the full height.

So the above-water height is

8000 \cdot \tfrac{1}{2} = 4000

feet.

The ocean depth at the base is the submerged height,

8000 - 4000 = 4000

feet.

Thus, the correct answer is A.

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