Heights and Distance – Objective Type Questions

Q 1 :

The length of the shadow of a tower on the plane ground is $\sqrt{3}$ times the height of the tower. The angle of elevation of the Sun is:

$30 °$
$45 °$
$60 °$
$90 °$

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(1)

Important points to understand in this question:

When Sun is behind the tower, AB, AC (on the ground) is the shadow of the tower.
The angle of elevation suggests that the point of observation is “upwards”.

Let h unit be the height of the tower, therefore $\sqrt{3} h$ is the length of the shadow of the tower and θ be angle of elevation of the Sun.

$I n ∆ B A C t a n θ = \frac{A B}{A C} = \frac{h}{\sqrt{3} h} \Rightarrow t a n θ = \frac{1}{\sqrt{3}} = t a n 30 ° \Rightarrow θ = 30 ° \Rightarrow Angle of elevation of the Sun is 30 °$

Q 2 :

If a pole 6 m high casts a shadow $2 \sqrt{3} m$ long on the ground, then Sun’s elevation is:

$60 °$
$45 °$
$30 °$
$90 °$

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(1)

Let AB be the pole of height 6 m. Let θ be the angle of elevation of Sun and BC be the shadow of the pole. Then,

$∠ A C B = θ, A B = 6 m a n d B C = 2 \sqrt 3 m I n △ A B C, r i g h t a n g l e d a t B,$

$\tan C = A B / B C \Rightarrow \tan θ = 6 / 2 \sqrt 3 = \sqrt 3$

$\Rightarrow t a n θ = t a n 60^{\circ} \Rightarrow θ = 60^{\circ}$

Q 3 :

A ladder makes an angle of $60 °$ with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder (in metres) is:

$\frac{4}{\sqrt{3}}$
$4 \sqrt{3}$
$2 \sqrt{2}$
4

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(4)

Let BC be wall and AC be the ladder placed against the wall. Then,

$A B = 2 m a n d ∠ B A C = 60 ° . I n △ A B C, w e h a v e$

$\cos 60 ° = \frac{A B}{A C} \Rightarrow \frac{1}{2} = \frac{2}{A C} \Rightarrow A C = 4 S o, t h e l e n g t h o f l a d d e r i s 4 m .$

Q 4 :

A clinometer is used to determine the height of a vertical cliff (an instrument used for measuring the angle or elevation of slopes). The angle of elevation is measured from 2 points (A and B respectively), which are 100 m and 200 m away from the base of the cliff.
At A and B, which of the following scenarios are possible?

(i) The angles of elevation from A and B to the top of the cliff are the same.
(ii) The angle of elevation from A is greater than the angle of elevation at B.
(iii) The angle of elevation at A is twice the angle of elevation at B.
(iv) B is farther from the cliff than A

(ii) only
(ii) and (iv) only
(i) and (iii) only
(ii), (iii) and (iv) only

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(2)

(i) Cannot be true (as angle of elevation will differ by the distance of the point to the base of the cliff).

(ii) If the angle of elevation at A is greater than at B, it indicates that A is closer to the cliff (which is suggested by the data).

$(i i i) L e t C D b e t h e v e r t i c a l c l i f f . T h e n, ∠ C A D = θ a n d ∠ C B D = ϕ . I n ∆ A C D, w e h a v e : t a n θ = \frac{C D}{A C} \Rightarrow t a n θ = \frac{h}{100} \dots (i) I n ∆ B C D, w e h a v e : t a n ϕ = \frac{C D}{B C} \Rightarrow t a n ϕ = \frac{h}{200} \dots (i i) F r o m (i) a n d (i i) : t a n θ = 2 t a n ϕ S o, θ \neq 2 ϕ . T h u s, s t a t e m e n t (i i i) i s n o t p o s s i b l e .$

(iv) B is farther (as suggested by the data).

Topic Question Set

Q 1 :

The length of the shadow of a tower on the plane ground is $\sqrt{3}$ times the height of the tower. The angle of elevation of the Sun is:

Q 2 :

If a pole 6 m high casts a shadow $2 \sqrt{3} m$ long on the ground, then Sun’s elevation is:

Q 3 :

A ladder makes an angle of $60 °$ with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder (in metres) is:

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