# A Cone Half Full

Consider a right circular cone with diameter of 12 cm for the base and altitude of 16 cm. If the cone is held with its vertex down and water placed into it until it is half full by volume, what is the depth of the water?

An aside, not really part of the problem:
How would you generate this image?

Hint for the problem?

# Discussion

There is a unique result for this problem. Can you find more than one way to solve for the result? Some possibilities:

1. Experimentally. Have students fill a cone (perhaps using rice rather than water), empty the contents into a beaker tomeasure the full volume, take half of it, and put half back in the cone. Measure the height. This might be a workable approach with younger students.

2. Calculate the full volume and solve for the height and radius of a cone with half the volume.

4. Find the depth x when a cone of radius r and height h is half full of water? Use this general solution to find the solution for this particular cone.

5. Let k range from 0 to 1. Plot the depth of water x as k goes from 0 to 1 for this cone. That is, find x = f(k) and graph it. This could be done either by deriving a function, or experimentally by filling the cone one cm at a time and measuring. Other graphs might be used.

x

k

Graph of x = f(k)