A hole is drilled through the center of a sphere. The cylinder-with-caps is removed. The length of the removed cylinder (it also has caps on it which do not count for the length) is 6 inches. What is the volume of the remaining solid?
There are two ways to do this problem.
- Here is the solution using calculus:
- Here is a solution which you may consider cheating. The very asking of the question implies that the answer can be determined from the data given. Hence we can CHOOSE an instance of the problem and KNOW that our solution for this instance is always the solution. We choose to have a cylinder of radius 0 (so its just is a line of length 6). Hence the answer is the Volume of a Sphere that is 6 inches in diameter: (4/3)(π)33=36π.
- Minor one: Deriving the volume of a sphere itself requires calculus so I didn't really get around that issue. However, the Volume of a sphere is well known so I think this is a quibble. (Does anyone know a non-calculus proof for the formula for the the volume of a sphere?)
- Major one: We used the fact that the answer can be determined from the data to find the answer. Is this appropriate?
- If you put this on an exam what would you do if a student had this solution? Reward them for thinking outside the box or penalize them for not showing they know calculus?
- What if it was on a mathematics competition?
- Best solution might be to make it a multiple choice question so they do not need to show how they did it. Those that think of the clever solution are rewarded by spending less time on it--- unless it took them a long time to think of the clever solution. Those that do it via calculus also get it right. You might want to make one of the choices Cannot be determined from the data given.