Question

A channel $27m$ wide falls at a right angle to another channel $64m$ wide. The greatest length of a log that can be floated along this system of channel is

• A. 110
• B. 125
• C. 100
• D. 120

Answer 1

The correct option is B 125

From figure it clear that the length of the log
$l=27\sec \theta +64\text{cosec}\theta$
$\Rightarrow \frac{d\ell }{d\theta }=27\sec \theta \tan \theta -64\text{cosec}\theta \cot \theta$
$=\frac{27{\sin }^3\theta -64{\cos }^3\theta }{\sin \theta \cos \theta }$

For maximum or minimum value of $l,d\theta =0$
$\Rightarrow {\tan }^3\theta =\frac{64}{27}=\frac{4^3}{3^3}$
$\Rightarrow \tan \theta =\frac{4}{3}$
$\Rightarrow \frac{\sin \theta }{4}=\frac{\cos \theta }{3}=\frac{1}{5}$
$\Rightarrow \sin \theta =\frac{4}{5},\cos \theta =\frac{3}{5}$.

Clearly, $\frac{d^2\ell }{d{\theta }^2}<0$ for $\tan \theta =\frac{4}{3}$
Hence maximum $\left(l\right)=27\times \frac{5}{3}+64\times \frac{5}{4}=125\text{m}$