Maximum Volume of a Cone

Click here for a problem description

Our objective is to find the value of t for which the volume of the resulting cone is a maximum. While this is a pretty standard Calculus problem - the challenge is to see if we can do it without Calculus and use the AM- GM inequality

That follows fairly easily from the structure of the problem.

. By Pythagoras which leads us to .


Then by the AM- GM inequality: with equality iff:

[Note: This use of the A.M.-G.M. inequality does not give us a function on the Right Hand Side of the inequality that is a constant. Therefore, even though we can find a value of x for which the two functions are tangent at that point, we can not argue that the function on the RHS is larger for all values of x.]

It follows that: and

The maximum volume is given by:

The value of t for which the volume is a maximum is: