Problem:
A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet?
Answer Choices:
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Solution:
The portion of the mountain that is above the water forms a cone that is similar to the entire mountain. The ratio of the volumes of the cones is the cube of the ratio of their heights. Let be the depth of the ocean, in feet. Then the height of the mountain above the water is feet, and
Taking cube roots on both sides gives
from which , and .
The problems on this page are the property of the MAA's American Mathematics Competitions