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?

$4000$

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

$6000$

$6400$

$7000$

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.

2008 AMC 12B Problem 11

Solution:

Problem 11 in Other Years