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\vspace*{-10mm}
- 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
$$
\tan^3\alpha =\frac a b \;.
$$
- 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
$$
w= \left(\frac {a -b \tan^3 \beta} {1 - \tan^2 \beta} \right)
\cos \beta \;.
$$