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.
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 $AQ$.
$\blacksquare$
Historical Note
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.
Sources
- 1714: Barbara Sidway: Question 36 (The Ladies' Diary )
- 1817: Thomas Leybourn: The Mathematical Questions Proposed in the Ladies' Diary and Their Original Answers: Geometrical Problems relating to Maxima and Minima
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The Ladies' Diary or Woman's Almanac, $\text {1704}$ – $\text {1841}$: $139$