Question : The distance between the two pillars is 120 metres. The height of one pillar is three times the other. The angles of elevation of their tops from the midpoint of the line connecting their feet are complementary to each other. The height (in metres) of the taller pillar is:
Option 1: 34.64
Option 2: 51.96
Option 3: 69.28
Option 4: 103.92
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Correct Answer: 103.92
Solution : Given: The distance between the two pillars is 120 metres. The height of one pillar is three times the other. Let the height of the taller pillar be $H_{2}$ and the height of the smaller pillar be $H_{1}$ and the angle of elevation to the top of the taller pillar is $(90°– \theta)$ and for the smaller pillar is $\theta$ from the midpoint of 120 metres. $H_{2}=3x$ $H_{1}=x$ $\tan\theta = \frac{\text{Height}}{\text{Base}}$ $⇒\tan(90°-\theta) = \frac{H_{2}}{60}$ $⇒\cot\theta = \frac{3x}{60}$ (equation 1) $⇒\tan\theta = \frac{H_{1}}{60}$ $⇒\tan\theta = \frac{x}{60}$ (equation 2) Now divide equation 2 by equation 1, we get, $\frac{\tan\ \theta}{\cot\ \theta}= \frac{60x}{60×3x}$ $⇒\frac{\tan\ \theta}{\cot\ \theta}= \frac{1}{3}$ $⇒ \tan^{2} \theta= \frac{1}{3}$ $⇒\tan\ \theta= \frac{1}{\sqrt{3}}=\tan 30°$ $\therefore\theta = 30°$ Now, the height of the taller pillar is $\tan(90°-\theta) = \frac{H_{2}}{60}$ $⇒\tan(90°- 30°) = \frac{H_{2}}{60}$ $⇒\tan60° = \frac{H_{2}}{60}$ $\therefore H_{2} = 60\sqrt{3}= 103.92$ m Hence, the correct answer is 103.92.
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Question : Two posts are $x$ metres apart and the height of one is double that of the other. If, from the midpoint of the line joining their feet, an observer finds the angular elevation of their tops to be complementary, then the height (in metres) of the shorter posts is:
Option 1: $\frac{x}{2\sqrt{2}}$
Option 2: $\frac{x}{4}$
Option 3: $x\sqrt{2}$
Option 4: $\frac{x}{\sqrt{2}}$
Question : The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. The height of the tower is:
Option 1: 4 metres
Option 2: 7 metres
Option 3: 9 metres
Option 4: 6 metres
Question : If the angle of elevation of the top of a pillar from the ground level is raised from 30° to 60°, the length of the shadow of a pillar of height $50\sqrt{3}$ metres will be decreased by:
Option 1: 60 metres
Option 2: 75 metres
Option 3: 100 metres
Option 4: 50 metres
Question : The angle of elevation of the top of an unfinished pillar at a point 150 metres from its base is 30°. The height (in metres) that the pillar raised so that its angle of elevation at the same point may be 45°, is:
Option 1: 63.4
Option 2: 86.6
Option 3: 126.8
Option 4: 173.2
Question : If the angle of elevation of the sun decreases from $45^\circ$ to $30^\circ$, then the length of the shadow of a pillar increases by 60 m. The height of the pillar is:
Option 1: $60(\sqrt{3}+1)$ metres
Option 2: $30(\sqrt{3}–1)$ metres
Option 3: $30(\sqrt{3}+1)$ metres
Option 4: $60(\sqrt{3}–1)$ metres
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