Longest cylinder of specified radius in a given cuboid Find the maximum height (in exact value) of a cylinder of radius $x$ so that it can completely place into a $100 cm \times 60 cm \times50 cm$ cuboid.
This question comes from http://hk.knowledge.yahoo.com/question/question?qid=7012072800395.
I know that this question is equivalent to two times of the maximum height (in exact value) of a right cone of radius $x$ so that it can completely place into a $50 cm \times$ $30 cm \times 25 cm$ cuboid whose the apex of the right cone is placed at the corner of the cuboid, but I still have no idea until now.
 A: Here's a possibly-wrong approach ...
Center a "short" cylinder at the cuboid's center, oriented along a diagonal. Elongate the cylinder ---in each direction--- until it collides with a face, say the "top" (and, at the other end, the "bottom"). Letting the cylinder scrape against the top (and bottom) face(s), continue elongating the cylinder ---such that the projection of its axis into the top face coincides with the face's diagonal--- until it collides with a second (pair of) face(s). Finally, letting the cylinder scrape those faces, elongate in the only dimension allowable until it collides with the final (pair of) face(s).
Symbolically ...
Let the cuboid have dimensions $2a$, $2b$, $2c$; center it at the origin, and let its edges be axis-aligned. Let the cylinder have radius $s$. (I use "$r$" below for a different parameter.)
Taking $P = p \; ( a, b, c )$ to be the center of one end of the "short" cylinder, we elongate the cylinder (that is, we increase $p$) until it collides with the walls $x=\pm a$; this happens when
$$P_x + s \frac{P_x}{|P|} = a \qquad (1)$$
Let $p_\star$ be the value of $p$ solving $(1)$.
Now, let $Q := p_\star \; ( a, b, c ) + q \; ( 0, b, c )$ take over as the center of the end of the cylinder; we elongate until the cylinder hits the walls $y=\pm b$:
$$Q_y + s \frac{Q_y}{|Q|} = b \qquad (2)$$
Writing $q_\star$ for the appropriate value of $q$ solving $(2)$, we finish with cylindrical endpoint $R := p_\star \; ( a, b, c ) + q_\star\; ( 0, b, c ) + r ( 0, 0, c )$ until
$$R_z + s \frac{R_z}{|R|} = c \qquad (3)$$
when $r = r_\star$. Then
$$\ell = 2|R| = 2\sqrt{\; p_\star^2 \; a^2 + \left( p_\star + q_\star \right)^2 \; b^2 + \left( p_\star + q_\star + r_\star \right)^2 \; c^2 \; }$$
may (or may not) be the length of the longest cylinder in the cuboid. At least, it should be a "local maximum".
As for solving $(1)$, $(2)$, $(3)$ ...
For $(1)$, defining $d := \sqrt{a^2+b^2+c^2}$, we have
$$p a + s \frac{p a}{p d} = a \qquad \to \qquad p_\star = 1 - \frac{s}{d}$$
For $(2)$, we have
$$(p_\star+q) b + \frac{s(p_\star + q )b}{\sqrt{p_\star^2 a^2 + (p_\star + q )^2(b^2+c^2)}} = b$$
$$\to \qquad s(p_\star + q ) = \left(1-p_\star-q\right)\;\sqrt{p_\star^2 a^2 + (p_\star + q )^2(b^2+c^2)}$$
$$\to \qquad s^2 (p_\star + q )^2 = \left(1-p_\star-q\right)^2\;\left( p_\star^2 a^2 + (p_\star + q )^2(b^2+c^2) \right)$$
which makes $q_\star$ the root of a quartic polynomial. The same is true for $r_\star$. I'll leave the sorting-out of those roots to the reader.
A: Does the $x$ axis-aligned? Would it possible to have $x=0$? The only thing want to mention is the maximum length of a line within the bounded space would be the diagonal, based on the metric property.
A: A beginning:
Let $a_i>0$ $\>(1\leq i\leq 3)$ be the dimensions of the box. Then we are looking for a unit vector ${\bf u}=(u_1,u_2,u_3)$ in the first octant and a length $\ell>0$ such that
$$\ell u_i+2 x\sqrt{1-u_i^2}=a_i\qquad(1\leq i\leq 3)\ .$$
When $x$ is small compared to the dimensions of the box one might begin with
$$\ell^{(0)}:=d:=\sqrt{a_1^2+a_2^2+a_3^2}\ ,\qquad u_i^{(0)}:={a_i\over d}\quad(1\leq i\leq3)$$
and do a few Newton iterations in order to obtain an approximate solution.
