JEE Main 2016 (Offline)Question 1
A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point $A$ on the path, he observes that the angle of elevation of the top of the pillar is $30^\circ$. After walking for $10$ minutes from $A$ in the same direction, at a point $B$, he observes that the angle of elevation of the top of the pillar is $60^\circ$. Then the time taken (in minutes) by him, from $B$ to reach the pillar, is
Trigonometric Ratio & Identities
Solution
Concept. Express both distances from the pillar in terms of its height $h$; time is proportional to distance. Formulas used. (1) Distance from the foot $=h\cot\theta$ for elevation $\theta$Step 1 - The picture.Step 2 - Distances. By (1): $A$ is at $h\cot30^\circ=h\sqrt3$ and $B$ is at $h\cot60^\circ=\dfrac{h}{\sqrt3}$. Step 3 - Compare. $$AB=h\sqrt3-\frac{h}{\sqrt3}=\frac{2h}{\sqrt3},\qquad B\text{ to foot}=\frac{h}{\sqrt3}=\frac12\,AB.$$ Step 4 - Time. At uniform speed, half the distance takes half the time: $\frac{10}{2}=5$ minutes. Answer: (B)