Maximum length of pencil in a pencil case
What is the maximum length of an unsharpened, cylindrical pencil inside an
empty rectangular pencil box?
Or, in a rectangular cuboid of dimensions $x \times y \times z$, what is
the maximum possible height of a right cylinder of radius r, that can be
inscribed within? Assume $x \ge y \ge z \ge 2r \ge 0$.
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