2008 AMC 12B Problem 11

Problem 11 of 25IntermediateGeometry

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?

Show solution

Solution

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.