question_answer

The length of the shadow of a person s cm tall when the angle of elevation of the sun is \[\alpha \] is p cm. It is q cm, when the angle of elevation of the sun is \[\beta .\]Which one of the following is correct, when \[\beta =3\,\,\alpha ?\]

A)  \[p-q=s\left( \frac{\tan \alpha -\tan 3\alpha }{\tan 3\alpha \tan \alpha } \right)\]

B)  \[p-q=s\left( \frac{\tan 3\alpha -\tan \alpha }{3\tan 3\alpha tan\alpha } \right)\]

C)  \[p-q=s\left( \frac{\tan 3\alpha -\tan \alpha }{\tan 3\alpha \tan \alpha } \right)\]

D)  \[p-q=s\left( \frac{\tan 2\alpha }{\tan 3\alpha \tan \alpha } \right)\]

Correct Answer: C

Solution :

In \[\Delta ABC,\]

\[\tan \alpha =\frac{s}{p}\]\[\Rightarrow \]\[p=\frac{s}{\tan \alpha }\]                     ?(i)

In \[\Delta BDC,\]

\[\tan \beta =\frac{s}{q}\]

\[\Rightarrow \]   \[q=\frac{s}{\tan 3\alpha }\]                   ?(ii)

On subtracting Eq. (ii) from Eq. (i), we get

\[p-q=\frac{s}{\tan \alpha }-\frac{s}{\tan 3\alpha }\]

\[=s\left( \frac{\tan 3\alpha -tan\alpha }{\tan \alpha \tan 3\alpha } \right)\]