Ladies' Diary/Largest Cylinder from given Cone

Puzzle

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

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

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.

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 $AQ$.

$\blacksquare$

This often-cited problem was raised by Mrs. Barbara Sidway, about whom little is known.

David Wells informs us in his Curious and Interesting Puzzles of $1992$ that the solution as given here is "something of a cheat", as it refers the reader to to one of Mr. Hutton's texbooks.

This is puzzling, however, as this problem is cited as appearing in $1714$, and Charles Hutton was not born until $1737$.

It was often the case that questions to The Ladies' Diary were taken from already-published sources.

1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The Ladies' Diary or Woman's Almanac, $\text {1704}$ – $\text {1841}$: $139$