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?

$4000$

$2000(4 - \sqrt{2})$

$6000$

$6400$

$7000$

答案：A

知识点：圆锥相似长度、面积与体积的缩放关系

难度评级：1570

解答：

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.

2008 AMC 12B 第 11 题

答案：A

难度评级：1570

解答：

其他年份的第 11 题