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: