2008 AMC 12B Problem 11

Question

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

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

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

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.

Problem 11 in other years