# RD Sharma Solutions for CBSE Class 10 Mathematics chapter 12 - Heights and Distances

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## Chapter 12 - Some Applications of Trigonometry Excercise Ex. 12.1

A ladder 15 metres long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, find the height of the wall.

^{o}to 60

^{o}as he walks towards the building. Find the distance he walked towards the building.

^{o}. Find the height of the tower.

^{o}and 60

^{o}respectively. Find the height of the tower.

Let BC be the building, AB be the transmission tower, and D be the point on ground from where elevation angles are to be measured.

^{o}and 45

^{o}respectively. Find the height of the multistoried building and the distance between the two buildings.

^{o}and from the same point the angle of elevation of the top of the pedestal is 45

^{o}. Find the height of the pedestal.

Let AB be the statue, BC be the pedestal and D be the point on ground from where elevation angles are to be measured.

^{o}. From another point 20 m away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is 30

^{o}. Find the height of the tower and the width of the canal.

^{o}and the angle of depression of its foot is 45

^{o}. Determine the height of the tower.

^{o}and 45

^{o}. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Let AB be the lighthouse and the two ships be at point C and D respectively.

^{o}and the angle of elevation of the top of the tower from the foot of the building is 60

^{o}. If the tower is 50 m high, find the height of the building.

^{o}and 45

^{o}respectively. If bridge is at the height of 30 m from the banks, find the width of the river.

^{o}and 30

^{o}, respectively. Find the height of poles and the distance of the point from the poles.

An aeroplane is flying at a height of 210 m. Flying at this height at some instant the angles of depression of two points in a line in opposite directions on both the banks of the river are 45° and 60°. Find the width of the river.

The angle of elevation of the top of a chimney from the top of the tower is 60° and the angle of depression of the foot of the chimney from the top of the tower is 30°. If the height of the tower is 40, find the height of the chimney. According to pollution control norms, the minimum height of a smoke emitting chimney should be 100m. State if the height of the above mentioned chimney meets the pollution norms. What value is discussed in this question?

Let AC = h be the height of the chimney.

Height of the tower = DE = BC = 40 m

In ∆ABE,

∴AB = BE√3….(i)

In ∆CBE,

tan 30° =

Substituting BE in (i),

AB = 40√3 × √3

= 120 m

Height of the chimney = AB + BC = 120 + 40 = 160 m

Yes, the height of the chimney meets the pollution control norms.

Two ships are there in the sea on either side of a light house in such away that the ships and the light house are in the same straight line. The angles of depression of two ships are observed from the top of the light house are 60° and 45° respectively. If the height of the light house is 200 m, find the distance between the two ships.

Let the ships be at B and C.

In D ABD,

∴ BD = 200 m

In D ADC,

Distance between the two ships = BC = BD + DC

The horizontal distance between two poles is 15 m. The angle of depression of the top of the first pole as seen from the top of the second pole is 30°. If the height of the second pole is 24 m, find the height of the first pole.

Here m∠CAB = m∠FEB = 30°.

Let BC = h m, AC = x m

In D ADE,

In D BAC,

Height of the second pole is 15.34 m

The angles of depression of two ships from the top of a light house and on the same side of it are found to be 45^{o} and 30^{o} respectively. If the ships are 200 m apart, find the height of the light house.

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. Prove that the height of the tower is 6 m.

Let AQ be the tower and R, S respectively be the points which are 4m, 9m away from base of tower.

As the height can not be negative, the height of the tower is 6 m.

From the top of building AB, 60 m high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30^{o} and 60^{o} respectively. Find

(i) the horizontal distance between AB and CD.

(ii) the height of the lamp post.

(iii) the difference between the heights of the building and the lamp post.

ΔA moving boat is observed from the top of a 150m high cliff moving away from the cliff. The angle of depression of the boat changes from 60˚ to 45˚ in 2 minutes. Find the speed of the boat in m /h.

Let AB be the cliff, so AB=150m.

C and D are positions of the boat.

DC is the distance covered in 2 min.

∠ACB = 60^{o} and ∠ADB = 45^{o}

∠ABC = 90^{o}

In ΔABC,

tan(∠ACB)=

In ΔABD,

tan(∠ADB)=

So, DC=BD - BC

=

Now,

From the top of a 120 m high tower, a man observes two cars on the opposite sides of the tower and in straight line with the base of tower with angles of depression as 60˚ and 45˚. Find the distance between the cars.

AB is the tower.

DC is the distance between cars.

AB=120m

In ΔABC,

tan(∠ACB) =

In ΔABD,

tan(∠ADB) =

So, DC=BD+BC

Two points A and B are on the same side of a tower and in the same straight line with its base. The angles of depression of these points from the top of the tower are 60˚ and 45˚ respectively. If the height of the tower is 15 m, then find the distance between these points.

Let CD be the tower.

So CD =15m

AB is the distance between the points.

∠CAD = 60^{o} and ∠CBD = 45^{o}

∠ADC = 90^{o}

In ΔADC,

tan(∠CAD)=

In ΔCBD,

tan(∠CBD)=

So AB=BD - AD

A fire in a building B is reported on telephone to two fire stations P and Q, 20 km apart from each other on a straight road. P observes that the fire is at an angle of 60^{o} to the road and Q observes that it is at an angle of 45^{o} to the road. Which station should send its team and how much will this team have to travel?

Now, in triangle APB,

sin 60^{o} = AB/ BP

√3/2 = h/ BP

This gives

h = 14.64 km

A man standing on the deck of a ship, which is 8 m above water level. He observes the angle of elevation of the top of a hill as 60^{o} and the angle of depression of the base of the hill as 30^{o}. Calculate the distance of the hill from the ship and the height of the hill.

The angle of elevation of an aeroplane from a point on the ground is 45^{o}. After a flight of 15 seconds, the elevation changes to 30^{o}. If the aeroplane is flying at a height of 3000 metres, find the speed of the aeroplane.

The angle of elevation of a stationery cloud from a point 2500 m above a lake is 15^{o} and the angle of depression of its reflection in lake is 45^{o}. What is the height of the cloud above the lake level? (Use tan 15^{o} = 0.268)

From the top of a tower h metre high, the angles of depression of two objects, which are in the line with the foot of the tower are α and β (β > α). Find the distance between the two objects.

A window of a house is h metre above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be a and b respectively. Prove that the height of the house is h (1 + tan α cot β) metres.

The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m vertically above the lower window. At certain instant the angles of elevation of a balloon from these window are observed to be 60° and 30° respectively. Find the height of the balloon above the ground.

## Chapter 12 - Some Applications of Trigonometry Excercise 12.41

If the altitude of the sum is at 60°, then the height of the vertical tower that will cast a shadow of length 30 m is

If the angles of elevation of a tower from two points distant a and b (a > b) from its foot and in the same straight line from it are 30° and 60°, then the height of the tower is

If the angles of elevation of the top of a tower from two points distant a and b from the base and in the same straight line with it are complementary, then the height of the tower is

From a light house the angles of depression of two ships on opposite sides of the light house are observed to be 30° and 45°. If the height of the light house is h metres, the distance between the ships is

The angle of elevation of the top of a tower standing on a horizontal plane from a point A is α. After walking a distance d towards the foot of the tower the angle of elevation is found to be β. The height of the tower is

The tops of two poles of height 20 m and 14 m are connected by a wire. If the wire makes an angle of 30° with horizontal, then the length of the wire is

(a) 12 m

(b) 10 m

(c) 8 m

(d) 6 m

Wire BD

ED || AC

So, EA = DC and ED = AC

EA = 14

AB = EA + EB

20 = 14 + EB

EB = 6

So, the correct option is (a).

From the top of a cliff 25 m high the angle of elevation of a tower is found to be equal to the angle of depression of the foot of the tower. The height of the tower is

(a) 25 m

(b) 50 m

(c) 75 m

(d) 100 m

The angles of depression of two ship from the top of a light house are 45° and 30° towards east. If the ships are 100 m apart, the height of the light house is

## Chapter 12 - Some Applications of Trigonometry Excercise 12.42

If the angle of elevation of a cloud from point 200 m above a lake is 30° and the angle depression of its reflection in the lake is 60°, then the height of the cloud above the lake, is

(a) 200 m

(b) 500 m

(c) 30 m

(d) 400 m

The height of a tower is 100 m. When the angle of elevation of the sun changes from 30° to 45°, the shadow of the tower becomes x metres less. The value of x is

(a) 100 m

Two persons are a metres apart and the height of one is double that of the other. If from the middle point of the line joining their feet, an observer finds the angular elevation of their tops to be complementary, then the height of the shorter post is

If height of one person is x then height of another one is 2x. Also If angle of elevation of one is θ then for another it is 90 - θ.

AB = a

C is mid point.

So, the correct option is (d).

The angle of elevation of a cloud from a point h metre above a lake is θ. The angle of depression of its reflection in the lake is 45°. The height of the cloud is

(a) h tan (45° + θ)

(b) h cot (45° - θ)

(c) h tan (45° - θ)

(d) h cot (45° + θ)

A tower subtends an angle of 30° at a point on the same level as its foot. At a second point h metres above the first, the depression of the foot of the tower is 60°. The height of the tower is

It is found that on walking x metres towards a chimney in a horizontal line through its base, the elevation of its top changes from 30° to 60°. The height of the chimney is

The length of the shadow of a tower standing on level ground is found to be 2x metres longer when the sun's elevation is 30° than when it was 45°. The height of the tower in metres is

Two poles are 'a' metres apart and the height of one is double of the other . If from the middle point of the line joining their feet an observer finds the angular elevations of their tops to be complementary, then the height of the smaller is

If height of one pole is x then height of the other one is 2x. Also If the angle of elevation of one is θ then for the other it is

90 - θ.

AB = a

C is mid point.

So, the correct option is (b).

The tops of two poles of height 16 m and 10 m are connected by a wire of length l metres. If the wire makes an angle 30° with the horizontal, then l =

(a) 26

(b) 16

(c) 12

(d) 10

EC || AB

Hence

EA = CB = 10

AD = AE + ED

ED = AD - AE

= 16 - 10 = 6

So, the correct option is (c).

If a 1.5 m tall girl stands at a distance of 3 m from a lamp-post and casts a shadow of length 4.5 m on the ground, then the height of the lamp-post is

(a) 1.5 m

(b) 2 m

(c) 2.5 m

(d) 2.8 m

## Chapter 12 - Some Applications of Trigonometry Excercise 12.43

The angle of depression of a car, standing on the ground, from the top of a 75 m tower, is 30°. The distance of the car from the base of the tower (in metres) is

A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, then the height of the wall is

The angle of depression of a car parked on the road from the top of a 150 m high tower is 30°. The distance of the car from the tower (in metres) is

The angle of elevation of the top of a tower at a point on the ground 50 m away from the foot of tower is 45°. Then the height of the tower (in metre) is

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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