(B) Let $PQ$ represent the pillar with height $h$.In $\Delta PAQ$,$\tan(30^o) = \frac{PQ}{AQ}$ $\Rightarrow \frac{1}{\sqrt{3}} = \frac{h}{AQ}$ $\Right

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(B) Let $PQ$ represent the pillar with height $h$.In $\Delta PAQ$,$\tan(30^o) = \frac{PQ}{AQ}$ $\Rightarrow \frac{1}{\sqrt{3}} = \frac{h}{AQ}$ $\Rightarrow AQ = h\sqrt{3}$.In $\Delta PQB$,$\tan(60^o) = \frac{PQ}{BQ}$ $\Rightarrow \sqrt{3} = \frac{h}{BQ}$ $\Rightarrow BQ = \frac{h}{\sqrt{3}}$.The distance covered in $10 \text{ minutes}$ is $AB = AQ - BQ = h\sqrt{3} - \frac{h}{\sqrt{3}} = \frac{3h - h}{\sqrt{3}} = \frac{2h}{\sqrt{3}}$.Since the speed is uniform,the time taken is proportional to the distance.Time taken to cover $AB = 10 \text{ minutes}$.Time taken to cover $BQ = \frac{BQ}{AB} \times 10 = \frac{h/\sqrt{3}}{2h/\sqrt{3}} \times 10 = \frac{1}{2} \times 10 = 5 \text{ minutes}$.

Class 11 Mathematics Trigonometrical Equations Height and Distance

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$A$ 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 $30^o$. After walking for $10 \text{ minutes}$ from $A$ in the same direction,at a point $B$,he observes that the angle of elevation of the top of the pillar is $60^o$. Then the time taken (in minutes) by him,from $B$ to reach the pillar,is:

JEE MAIN 2016 All JEE MAIN

A
$20$
B
$5$
C
$6$
D
$10$

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Trigonometrical Equations

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Height and Distance

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In the figure,$\theta_1+\theta_2=\frac{\pi}{2}$ and $\sqrt{3}(BE)=4(AB)$. If the area of $\triangle CAB$ is $2\sqrt{3}-3 \text{ unit}^2$,when $\frac{\theta_2}{\theta_1}$ is the largest,then the perimeter (in unit) of $\triangle CED$ is equal to $...........$.

JEE MAIN 2023

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An observer in a boat finds that the angle of elevation of the top of a tower standing on the top of a cliff is $60^\circ$ and that of the top of the cliff is $30^\circ$. If the height of the tower is $60 \ m$,then the height of the cliff is ... $m$.

Medium

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The horizontal distance between two towers is $60 \ m$. The angle of depression of the top of the first tower as seen from the top of the second tower is $30^\circ$. If the height of the second tower is $150 \ m$,then the height of the first tower is:

Medium

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$A$ tower $PQ$ stands on a horizontal ground with base $Q$ on the ground. The point $R$ divides the tower into two parts such that $QR = 15 \, m$. If from a point $A$ on the ground,the angle of elevation of $R$ is $60^{\circ}$ and the part $PR$ of the tower subtends an angle of $15^{\circ}$ at $A$,then the height of the tower is:

DifficultJEE MAIN 2022

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$A$ sphere with centre $O$ sits on the top of a pole as shown in the figure. An observer on the ground is at a distance $50 \ m$ from the foot of the pole. She notes that the angles of elevation from the observer to points $P$ and $Q$ on the sphere are $30^{\circ}$ and $60^{\circ}$,respectively. Then,the radius of the sphere in metres is

KVPY 2016

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In the figure,$\theta_1+\theta_2=\frac{\pi}{2}$ and $\sqrt{3}(BE)=4(AB)$. If the area of $\triangle CAB$ is $2\sqrt{3}-3 \text{ unit}^2$,when $\frac{\theta_2}{\theta_1}$ is the largest,then the perimeter (in unit) of $\triangle CED$ is equal to $...........$.

An observer in a boat finds that the angle of elevation of the top of a tower standing on the top of a cliff is $60^\circ$ and that of the top of the cliff is $30^\circ$. If the height of the tower is $60 \ m$,then the height of the cliff is ... $m$.

The horizontal distance between two towers is $60 \ m$. The angle of depression of the top of the first tower as seen from the top of the second tower is $30^\circ$. If the height of the second tower is $150 \ m$,then the height of the first tower is:

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