Ladies' Diary/Largest Cylinder from given Cone

Puzzle

 * From a given (right circular) cone to cut the greatest cylinder possible.

Solution
The maximum volume of the cylinder when its height is $\dfrac 2 3$ of the height of the cone.

Proof
It is assumed the base of the cylinder is on the base of the cone.

It remains to determine its height as a proportion of the height of the cone.


 * Largest-Cylinder-in-Cone.png

The volume of the cylinder is proportional to $DE^2 \times PQ$.

We have that $\dfrac {DE} {AP}$ is constant.

Therefore we are to ensure that $AP^2 \times PQ$ is a maximum.

That is, given any line $AQ$, find a point $P$ on it such that $AP^2 \times PQ$ is a maximum.

Let $AP$ be denoted $x$.

Let $AQ$ be denoted $L$.

Thus we need to make $x^2 \paren {L - x}$ a maximum.

That is the same as making $x^2 \paren {2 L - 2 x}$ a maximum.

But the latter is the product of $3$ factors whose sum is constant at $2 L$.

It will achieve that maximum when all $3$ are equal.

Therefore the maximum is when $x = 2 L - 2 x$ and $x = \dfrac {2 L} 3$.

So $AP$ is $\dfrac 2 3$ of $AQ$.