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An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt{3} k m$ km above it, is observed at an angle of elevation of $60^{o}$ from a point on the ground. If after five seconds, its angle of elevation from the same point is $30^{o}$ then the speed (in $k m / h r$ ) of the aeroplane is _____

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Important Questions on Heights and Distances

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

If the angles of elevation of the top of a tower from three collinear points

A, B

and

C

on a line leading to the foot of the tower are

30 °, 45 °

and

60 °

respectively, then the ratio

A B : B C

, is

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes

18 \min

for the angle of depression of the car to change from

30 °

45 °

, then the time taken (in

\min

) by the car to reach the foot of the tower is

HARD

Mathematics>Trigonometry>Heights and Distances>Heights and Distances

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^{°}

. 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^{°}

. Then the time taken (in minutes) by him, from

B

to reach the pillar, is

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

A B C

is a triangular park with

A B = A C = 100

metres. A vertical tower is situated at the mid-point of

B C

. If the angles of elevation of the top of the tower at,

A

and

B

are

{c o t}^{- 1} (3 \sqrt{2})

and

{c o s e c}^{- 1} (2 \sqrt{2})

respectively, then the height of the tower (in metres) is

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

A bird is sitting on the top of a vertical pole

20 m

high and its elevation from a point

O

on the ground is

45 °

. It flies off horizontally straight away from the point

O

. After one second, the elevation of the bird from

O

is reduced to

30 °

. Then the speed (in

m / s

) of the bird is

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

Two poles standing on a horizontal ground are of heights

5 m

and

10 m

respectively. The line joining their tops makes an angle of

15 °

with the ground. Then the distance

(

m)

between the poles, is

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

P Q R

is a triangular park with

P Q = P R = 200 m

. A T.V. tower stands at the mid-point of

Q R

. If the angles of elevation of the top of the tower at

P, Q

and

R

are respectively,

45 °, 30 °

and

30 °

, then the height of the tower (in

m

) is:

EASY

Mathematics>Trigonometry>Heights and Distances>Heights and Distances

A ladder $10 m$ long is leaning against a vertical wall. It makes an angle of $60 °$ with the ground. How far is the foot of the ladder from the wall?

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

A straight highway leads to the foot of a

50 m

tall tower. From the top of the tower the angle of depression of the two cars on the highway are

30 °

and

60 °

. Then the distance between the two cars is

HARD

Mathematics>Trigonometry>Heights and Distances>Heights and Distances

A sphere with centre $O$ sits atop 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 $30 °$ and $60 °$ , respectively. Then, the radius of the sphere in meters is

Question Image

EASY

Mathematics>Trigonometry>Heights and Distances>Heights and Distances

∆ XYZ

, the measure of angle

Y

90 °

. If

\sec X = \frac{17}{8}

, and

XY = 0.8 cm

, what is the length (in

cm

) of side

XZ

HARD

Mathematics>Trigonometry>Heights and Distances>Heights and Distances

If the angle of elevation of a cloud from a point

P

which is

25 m

above a lake be

30 °

and the angle of depression of reflection of the could in the lake from

P

60 °

, then the height of the cloud (in meters) from the surface of the lake is :

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

In triangle

ABC

measure of angle

B

90 °

. If

\sec A = \frac{25}{7}

and

AB = 14 cm

, what is the length (in

cm

) of side

BC

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

Two vertical poles of height,

20 m

and

80 m

stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is:

EASY

Mathematics>Trigonometry>Heights and Distances>Heights and Distances

Two poles of the height $12 m$ and $17 m$ stand vertically upright on a plain ground. If the distance between their feet is $12 m$ , find the distance between their tops.

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

P and Q are two points on the ground on either side of a pole. The angles of elevation of the top of the pole as observed from P and Q are

60^{o}

and

30^{o}

, respectively and the distance between them is

84 \sqrt{3} m

. What is the height (in

m

) of the pole?

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

X is $5 m$ tall and he notices that he casts a shadow that is $3 m$ long. He then measures that the shadow cast by his school building is $30 m$ long. How tall is the building?

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

Let a vertical tower

A B

have its end

A

on the level ground. Let

C

be the mid-point of

A B

and

P

be a point on the ground such that

A P = 2 A B

. If

∠ B P C = β

, then

\tan β

is equal to:

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

Let

10

vertical poles standing at equal distances on a straight line, subtend the same angle of elevation

α

at a point

O

on this line and all the poles are on the same side of

O

. If the height of the longest pole is

h

and the distance of the foot of the smallest pole from

O

a

; then the distance between two consecutive poles, is

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Mathematics>Trigonometry>Heights and Distances>Heights and Distances

Consider a triangular plot

A B C

with sides

A B = 7 m,

B C = 5 m

and

C A = 6 m .

A vertical lamp-post at the mid-point

D

A C

subtends an angle

30 °

B .

The height (in

m

) of the lamp-post is:

An aeroplane flying at a constant speed, parallel to the horizontal ground, 3 km km above it, is observed at an angle of elevation of 60o from a point on the ground. If after five seconds, its angle of elevation from the same point is 30o then the speed (in km/hr) of the aeroplane is _____

Important Questions on Heights and Distances

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