Problem source

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\begin{questionparts}
\item An attempt is made to move a rod of length $L$ from a corridor 
of width $a$ into a corridor of width~$b$, where $a \ne b.$ The corridors
meet at right angles, as shown in Figure 1 and the rod remains horizontal.
Show that if the attempt is to be successful then 
$$
L \le a \cosec {\alpha} + b \sec {\alpha} \;,
$$ 
where ${\alpha}$ satisfies 
$$
\tan^3\alpha =\frac a b \;.
$$

\item
An attempt is made to move a rectangular table-top,  of width $w$ and length $l$,
from one corridor to the other, as shown in the Figure 2. 
The table-top remains horizontal.
Show that if the attempt is to be successful then 
$$
l\le a \cosec {\beta} + b \sec {\beta}  -2w \cosec 2{\beta},
$$ 
where ${\beta}$ satisfies 
$$
w=  \left(\frac {a -b \tan^3 \beta} {1 - \tan^2 \beta} \right)
\cos \beta \;.
$$
\end{questionparts}