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

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4



Question 5

Solution 5

Question 6

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.

Solution 6

Question 7
Solution 7

Question 8
Solution 8


Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
A 1.6 m tall girl stands at a distance of 3.2 m from a lamp-post and casts a shadow of 4.8 m on the ground. Find the height of the lamp-post by using (i) trigonometric ratios (ii) property of similar triangles.
Solution 21




Question 22
A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30o to 60o as he walks towards the building. Find the distance he walked towards the building.
Solution 22
Question 23
The shadow of a tower standing on a level ground is found to be 40 m longer when Sun'saltitude is 30o. Find the height of the tower.
Solution 23

Question 24
From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45o and 60o respectively. Find the height of the tower.
Solution 24


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.

Question 25
The angles of depression of the top and bottom of 8 m tall building from the top of a multistoried building are 30o and 45o respectively. Find the height of the multistoried building and the distance between the two buildings.
Solution 25

Question 26
A statue, 1.6 m tall, stands on a top of pedestal, from a point on the ground, the angle of elevation of the top of statue is 60o and from the same point the angle of elevation of the top of the pedestal is 45o. Find the height of the pedestal.
Solution 26


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

Question 27
A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower the angle of elevation of the top of the tower is 60o. 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 30o. Find the height of the tower and the width of the canal.

Solution 27


Question 28
From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60o and the angle of depression of its foot is 45o. Determine the height of the tower.
Solution 28
Question 29
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30o and 45o. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
Solution 29


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

Question 30
The angle of elevation of the top of a building from the foot of the tower is 30o and the angle of elevation of the top of the tower from the foot of the building is 60o. If the tower is 50 m high, find the height of the building.
Solution 30
Question 31
From a point on a bridge across a river the angles of depression of the banks on opposite side of the river are 30o and 45o respectively. If bridge is at the height of 30 m from the banks, find the width of the river.
Solution 31

Question 32
Two poles of equal heights are standing opposite each other an either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60o and 30o, respectively. Find the height of poles and the distance of the point from the poles.
Solution 32
Question 33
Solution 33


Question 34
Solution 34

Question 35
Solution 35


Question 36
Solution 36


Question 37
Solution 37


Question 38
Solution 38


Question 39
Solution 39


Question 40

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.

 

Solution 40

 

 

  

 

 

Question 41

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?

Solution 41

 

 

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.

Question 42

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.

 

Solution 42

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

 

Question 43

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.

Solution 43

Here mCAB = mFEB = 30°.

Let BC = h m, AC = x m

In D ADE,

 

 

In D BAC,

 

 

Height of the second pole is 15.34 m

Question 44

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 45o and 30o respectively. If the ships are 200 m apart, find the height of the light house.

Solution 44



Question 45

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.

Solution 45



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.

Question 46
Solution 46

Question 47

Solution 47



Question 48

Solution 48

Question 49
Solution 49



Question 50

Solution 50



Question 51

Solution 51




Question 52

Solution 52



Question 53

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 30o and 60o 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.

Solution 53




Question 54

Solution 54




Question 55

Solution 55



Question 56

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.

Solution 56

  

 

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 = 60o and ADB = 45o

ABC = 90o

In ΔABC,

tan(ACB)=

In ΔABD,

tan(ADB)=

So, DC=BD - BC

  =

Now,

begin mathsize 12px style speed equals distance over time
equals fraction numerator begin display style fraction numerator 50 open parentheses 3 minus square root of 3 close parentheses over denominator 1000 end fraction end style km over denominator 2 cross times begin display style 1 over 60 end style hrs end fraction... take space square root of 3 equals 1.732
equals 1.9019 space km divided by hr
equals 1902 space straight m divided by hr end style

Question 57

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.

Solution 57

  

 

AB is the tower.

DC is the distance between cars.

AB=120m

In ΔABC,

tan(ACB) =

In ΔABD,

tan(ADB) =

So, DC=BD+BC

Question 58

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.

Solution 58

  

Let CD be the tower.

So CD =15m

AB is the distance between the points.

CAD = 60o and CBD = 45o

ADC = 90o

In ΔADC,

tan(CAD)=

In ΔCBD,

tan(CBD)=

So AB=BD - AD

Question 59

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 60o to the road and Q observes that it is at an angle of 45o to the road. Which station should send its team and how much will this team have to travel?

Solution 59




Now, in triangle APB,

sin 60o = AB/ BP

√3/2 = h/ BP

This gives

h = 14.64 km

Question 60

Solution 60



Question 61

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 60o and the angle of depression of the base of the hill as 30o. Calculate the distance of the hill from the ship and the height of the hill.

Solution 61

Question 62

Solution 62

Question 63

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

Solution 63



Question 64

Solution 64



Question 65

Solution 65

Question 66

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

Solution 66

Question 67

Solution 67



Question 68

Solution 68



Question 69

Solution 69




Question 70

Solution 70



Question 71

Solution 71



Question 72

Solution 72



Question 73

Solution 73



Question 74

Solution 74

Question 75

Solution 75



Question 76

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.

Solution 76

Question 77

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.

Solution 77

 

Question 78

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.

Solution 78

 

Chapter 12 - Some Applications of Trigonometry Excercise 12.41

Question 1

begin mathsize 12px style The space ratio space of space the space length space of space straight a space rod space and space its space shadow space is space 1 space colon space square root of 3. space The space angle space of space elevation space of space the space sum space is
open parentheses straight a close parentheses space 30 degree
open parentheses straight b close parentheses space 45 degree
open parentheses straight c close parentheses space 60 degree
open parentheses straight d close parentheses space 90 degree end style

Solution 1

begin mathsize 12px style straight theta space is space angle space of space elevation
given space AB over BC equals space fraction numerator 1 over denominator square root of 3 end fraction space space space space space space space..... open parentheses 1 close parentheses
We space know space tan space straight theta space equals space AB over BC space space space space space..... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
tan space straight theta space equals space fraction numerator 1 over denominator square root of 3 end fraction space space also space tan 30 degree space equals space fraction numerator 1 over denominator square root of 3 end fraction
Hence space straight theta space equals space 30 degree
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 2

begin mathsize 12px style If space the space angle space of space elevation space of space straight a space tower space from space straight a space distance space of space 100 space meters space from space its space foot space is space 60 degree comma space then space the space height space of space the space tower space is
open parentheses straight a close parentheses space 100 square root of 3 space straight m
open parentheses straight b close parentheses space fraction numerator 100 over denominator square root of 3 end fraction straight m
open parentheses straight c close parentheses space 50 square root of 3 space straight m
open parentheses straight d close parentheses space fraction numerator 200 over denominator square root of 3 end fraction straight m end style

Solution 2

begin mathsize 12px style Let space straight h space be space the space height space of space tower
tan space 60 degree space equals space straight h over BC
straight h space equals space BC space tan space 60 degree
space space space space equals space 100 square root of 3 space straight m
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 3

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

begin mathsize 12px style open parentheses straight a close parentheses space 30 square root of 3 space straight m
open parentheses straight b close parentheses space 15 space straight m
open parentheses straight c close parentheses space fraction numerator 30 over denominator square root of 3 end fraction straight m
open parentheses straight d close parentheses space 15 square root of 2 space straight m end style

Solution 3

begin mathsize 12px style AB space equals space Shadow space length
AB space equals space 30 space straight m
straight theta space equals space angle space of space elevation
straight theta space equals space 60 degree
tan space straight theta space equals space AC over AB
tan space 60 degree space equals space AC over 30
AC space equals space 30 square root of 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses tan 60 degree space equals space square root of 3 close parentheses
height space of space tower space equals space 30 square root of 3
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 4

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

begin mathsize 12px style open parentheses straight a close parentheses space square root of straight a space plus space straight b end root
open parentheses straight b close parentheses space square root of ab
open parentheses straight c close parentheses space square root of straight a space minus space straight b end root
open parentheses straight d close parentheses space square root of straight a over straight b end root end style

Solution 4

begin mathsize 12px style BD space equals space straight a
BC space equals space straight b
tan space 60 degree space equals space AB over BC space and space tan space 30 degree space equals space AB over BD
square root of 3 equals AB over straight b space space space space space space space space..... open parentheses 1 close parentheses space and space fraction numerator 1 over denominator square root of 3 end fraction equals AB over straight a space space space space space.... open parentheses 2 close parentheses
open parentheses 1 close parentheses space cross times space open parentheses 2 close parentheses
square root of 3 cross times fraction numerator 1 over denominator square root of 3 end fraction equals AB squared over ab
AB space equals space square root of ab
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 5

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

begin mathsize 12px style open parentheses straight a close parentheses space ab
open parentheses straight b close parentheses space square root of ab
open parentheses straight c close parentheses space straight a over straight b
open parentheses straight d close parentheses space square root of straight a over straight b end root end style

Solution 5

begin mathsize 12px style Angles space are space complementery. space Hence space if space one space is space straight theta space other space must space be space 90 minus space straight theta
AB space equals space straight a
AC space equals space straight b
Let space the space height space AD space equals space straight h
In space triangle ABD
tan space straight theta space equals space AD over AB space equals space straight h over straight a space space space space space space space.... open parentheses 1 close parentheses
In space triangle ACD
tan space open parentheses 90 space minus space straight theta close parentheses space equals space AD over AC space equals space straight h over straight b space space space space space.... open parentheses 2 close parentheses
We space know space tan space open parentheses 90 space minus space straight theta close parentheses space equals space cot space straight theta space and space cot space straight theta space equals space fraction numerator 1 over denominator tan space straight theta end fraction
so space tan open parentheses 90 space minus space straight theta close parentheses space equals space fraction numerator 1 over denominator tan space straight theta end fraction space space space space space space space space space space space space..... open parentheses 3 close parentheses
from space open parentheses 2 close parentheses space & space open parentheses 3 close parentheses
fraction numerator 1 over denominator tan space straight theta end fraction space equals space straight h over straight b
tan space straight theta space equals space straight b over straight h space space space space space space space space space space..... open parentheses 4 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 4 close parentheses
straight h over straight a equals space straight b over straight h
straight h squared space equals space ab
straight h space equals space square root of ab
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 6

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 

begin mathsize 12px style open parentheses straight a close parentheses space open parentheses square root of 3 space plus space 1 close parentheses straight h space metres
open parentheses straight b close parentheses space open parentheses square root of 3 space minus space 1 close parentheses straight h space metres
open parentheses straight c close parentheses space square root of 3 space straight h space metres
open parentheses straight d close parentheses space 1 space plus space open parentheses 1 space plus space fraction numerator 1 over denominator square root of 3 end fraction close parentheses straight h space metres end style

Solution 6

begin mathsize 12px style MN space vertical line vertical line space DC
Hence space angle MAC space equals space angle ACB
and space angle NAD space equals space angle ADB
so space angle straight c space equals space 45 degree space and space angle straight D space equals space 30 degree
tan space straight C space equals space AB over BC space space space space space space space space space space space space space space space space space space space space space space space space space space space tan space straight D space equals space AB over BD
tan space 45 degree space equals space straight h over BC space space space space space space space space space space space space space space space space space space tan space 30 degree space equals space straight h over BD
BC space equals space straight h space space space space space space space space..... open parentheses 1 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space BD space equals space square root of 3 straight h space space space space space space space space....... open parentheses 2 close parentheses
distance space between space ships space equals space DB space plus space BC
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight h space plus space square root of 3 straight h
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses 1 space plus space square root of 3 close parentheses straight h
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 7

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

begin mathsize 12px style open parentheses straight a close parentheses space fraction numerator straight d over denominator cot space straight alpha space plus space cot space straight beta end fraction
open parentheses straight b close parentheses space fraction numerator straight d over denominator cot space straight alpha space minus space cot space straight beta end fraction
open parentheses straight c close parentheses space fraction numerator straight d over denominator tan space straight beta space minus space tan space straight alpha end fraction
open parentheses straight d close parentheses space fraction numerator straight d over denominator tan space straight beta space plus space tan space straight alpha end fraction
end style

Solution 7

begin mathsize 12px style Let space CD space equals space straight h
tanα space equals space CD over CA space equals space fraction numerator straight h over denominator CB space plus space BA end fraction
tan space straight alpha space equals space fraction numerator straight h over denominator CB space plus space straight d end fraction
CB space plus space straight d space equals space fraction numerator straight h over denominator tan space straight alpha end fraction space space space space space space space space space space space space space space space..... open parentheses 1 close parentheses
tan space straight beta space equals space straight h over CB
CB space equals space fraction numerator straight h over denominator tan space straight beta end fraction space space space space space space space space space space space space.... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
fraction numerator straight h over denominator tan space straight beta end fraction plus space straight alpha space equals fraction numerator straight h over denominator tan space straight alpha end fraction
straight alpha space equals space hcot space straight alpha space minus space hcot space straight beta
space space space space space equals space straight h open parentheses cot space straight alpha space minus space cot space straight beta close parentheses
straight h equals space fraction numerator straight d over denominator cot space straight alpha space minus space cot space straight beta end fraction
So comma space the space correct space option space is space left parenthesis straight b right parenthesis.
end style

Question 8

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

Solution 8

Wire BD

ED || AC

So, EA = DC and ED = AC

EA = 14

AB = EA + EB

20 = 14 + EB

EB = 6

begin mathsize 12px style In space triangle EDB
sin space 30 degree space equals space EB over BD
sin space 30 degree space equals space fraction numerator EB over denominator length space of space wire end fraction
length space of space wire space equals space fraction numerator EB over denominator sin space 30 degree end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 6 over denominator open parentheses 1 divided by 2 close parentheses end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 12 space straight m end style

 So, the correct option is (a).

Question 9

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

Solution 9

begin mathsize 12px style Let space height space of space tower space is space straight h
straight theta space rightwards arrow space angle space of space elevation space of space tower space top
straight alpha space rightwards arrow space angle space of space depression space of space tower space foot
given space straight theta space equals space straight alpha
EC space vertical line vertical line space AB space and space EC space equals space AB space space space space space space space space space space space space space space space space space...... open parentheses 1 close parentheses
so space angle ECA space equals space angle CAB space space open parentheses Alternate space Interior space angle close parentheses
also space AD space parallel to space BC
Hence space AE space equals space BC space equals space 25 space straight m space space space space space space space space space..... open parentheses 2 close parentheses
Now space in space triangle ABC
tan space straight alpha space equals space BC over AB
tan space straight alpha space equals space 25 over AB space space space space space space space space space space space space.... open parentheses 3 close parentheses
Now space in space triangle space DEC
tan space straight theta space equals space ED over EC
tan space straight alpha space equals space fraction numerator AD space minus space AE over denominator EC end fraction
from space open parentheses 1 close parentheses comma space open parentheses 2 close parentheses comma space open parentheses 3 close parentheses
tan space straight alpha space equals space fraction numerator straight h space minus space 25 over denominator AB end fraction space and space tan space straight alpha space equals space 25 over AB
AB space tan space straight alpha space equals space straight h space minus space 25 space space space space space space space space space space space.... open parentheses 4 close parentheses
and space AB space tan space straight alpha space equals space 25 space space space space space space space space space space space space.. open parentheses 5 close parentheses
from space open parentheses 4 close parentheses space & space open parentheses 5 close parentheses
straight h space minus space 25 space equals space 25
straight h space equals space 50 space straight m
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 10

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

begin mathsize 12px style open parentheses straight a close parentheses space fraction numerator 50 over denominator square root of 3 space plus space 1 end fraction space straight m
open parentheses straight b close parentheses space fraction numerator 50 over denominator square root of 3 space minus space 1 end fraction space straight m
open parentheses straight c close parentheses space 50 open parentheses square root of 3 space minus space 1 close parentheses space straight m
open parentheses straight d close parentheses space 50 open parentheses square root of 3 space plus space 1 close parentheses space straight m end style

Solution 10

begin mathsize 12px style By space Alternate space interior space angle space property
angle straight B space equals space 45 degree space and space angle straight C space equals space 30 degree
Let space the space height space be space straight h
Now space tan space 30 degree space equals space straight h over AC space
AC space equals space square root of 3 straight h
AB space plus space BC space equals space square root of 3 straight h space space space space space space space space space space space space space space space space..... open parentheses 1 close parentheses
tan space 45 degree space equals space straight h over AB
AB space equals space straight h
AB space equals space straight h space space space space space space space space space space space space space....... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space and space open parentheses 2 close parentheses
straight h space plus space BC space equals space square root of 3 straight h
straight h space plus space 100 space equals space square root of 3 straight h
100 space equals space open parentheses square root of 3 space minus space 1 close parentheses straight h
straight h space equals space fraction numerator 100 over denominator square root of 3 minus space 1 end fraction cross times fraction numerator square root of 3 space plus space 1 over denominator square root of 3 space plus space 1 end fraction equals fraction numerator 100 open parentheses square root of 3 space plus space 1 close parentheses over denominator 3 space minus space 1 end fraction
straight h space equals space 50 open parentheses square root of 3 space plus space 1 close parentheses
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Chapter 12 - Some Applications of Trigonometry Excercise 12.42

Question 1

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

Solution 1

begin mathsize 12px style Let space cloud space is space at space height space straight h space from space lake comma space so space its space reflection space is space straight a space depth space straight h space from space the space lake.
OC space equals space OR space equals space straight h
AB space equals space OE space equals space 200 space straight m
In space triangle CEB
tan space 30 degree space equals space EC over EB
EB space equals space square root of 3 space EC space space space space space space space space space space space space space space space space space space space....... open parentheses 1 close parentheses
In space triangle BER
tan space 60 degree space equals space RE over EB
EB space equals space fraction numerator RE over denominator square root of 3 end fraction space space space space space space space space space space space space space space space space space space space space space space space...... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
square root of 3 EC space equals space fraction numerator RE over denominator square root of 3 end fraction
3 open parentheses OC space minus space OE close parentheses space equals space OR space plus space OE
3 open parentheses straight h space minus space 200 close parentheses space equals space straight h space plus space 200
3 straight h space minus space 600 space equals space straight h space plus space 200
2 straight h space equals space 800
straight h space equals space 400 space straight m
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 2

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

begin mathsize 12px style open parentheses straight b close parentheses space 100 square root of 3 space straight m
open parentheses straight c close parentheses space 100 open parentheses square root of 3 space minus space 1 close parentheses space straight m
open parentheses straight d close parentheses space fraction numerator 100 over denominator square root of 3 end fraction straight m end style

Solution 2

begin mathsize 12px style CD space equals space straight x
tan space 30 degree space equals space 100 over BD space and space tan space 45 degree space equals space AB over BC equals space 100 over BC
BD space equals space 100 square root of 3 space space space space space space and space space BC space equals space 100
BC space plus space CD space equals space 100 square root of 3
100 space plus space straight x space equals space 100 square root of 3
straight x space equals space 100 open parentheses square root of 3 space minus space 1 close parentheses space straight m
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 3

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

begin mathsize 12px style open parentheses straight a close parentheses space straight a over 4
open parentheses straight b close parentheses space fraction numerator straight a over denominator square root of 2 end fraction
open parentheses straight c close parentheses space straight a square root of 2 space
open parentheses straight d close parentheses space fraction numerator straight a over denominator 2 square root of 2 end fraction end style

Solution 3

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.

begin mathsize 12px style So space AC space equals space BC space equals space straight a over 2
tan space straight theta space equals space fraction numerator straight x over denominator open parentheses begin display style straight a over 2 end style close parentheses end fraction equals fraction numerator 2 straight x over denominator straight a end fraction space space space space space space space.... open parentheses 1 close parentheses
tan open parentheses 90 space minus space straight theta close parentheses space equals space fraction numerator 2 straight x over denominator open parentheses begin display style straight a over 2 end style close parentheses end fraction equals space fraction numerator 4 straight x over denominator straight a end fraction space space space space space space space space space space.... open parentheses 2 close parentheses
We space know comma space tan space open parentheses 90 space minus space straight theta close parentheses space equals space cotθ
space space space space space space space space space space space space space space space space space space space space space space tan space open parentheses 90 space minus space straight theta close parentheses space equals space fraction numerator 1 over denominator tan space straight theta end fraction
space space space space space space space space space space space space space space space space space space space space space space tan space straight theta space. space tan space open parentheses 90 space minus space straight theta close parentheses space equals space 1 space space space space space space space space space space space...... open parentheses 3 close parentheses
from space open parentheses 1 close parentheses comma space open parentheses 2 close parentheses comma space open parentheses 3 close parentheses
open parentheses fraction numerator 2 straight x over denominator straight a end fraction close parentheses open parentheses fraction numerator 4 straight x over denominator straight a end fraction close parentheses space equals space 1
fraction numerator 8 straight x squared over denominator straight a squared end fraction equals 1
straight x squared space equals space straight a squared over 8 space space space space space space space space space rightwards double arrow straight x equals fraction numerator straight a over denominator 2 square root of 2 end fraction end style

So, the correct option is (d).

Question 4

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)

Solution 4

begin mathsize 12px style Correct space option space colon space open parentheses straight a close parentheses
Let space height space of space cloud space from space lake space is space straight H. space So space reflection space is space at space straight H space depth space from space lake.
tan space straight theta space equals space DC over DA space space space space space space space space space space space space.... open parentheses 1 close parentheses
tan space 45 degree space equals space RD over DA space space space space space space space space space space..... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
fraction numerator DC over denominator tan space straight theta end fraction space equals space fraction numerator RD over denominator tan space 45 degree end fraction
fraction numerator OC space minus space OD over denominator tan space straight theta end fraction space equals space fraction numerator RO space plus space OD over denominator 1 end fraction
fraction numerator straight H space minus space straight h over denominator tan space straight theta end fraction space equals space straight H space plus space straight h
straight H space minus space straight h space equals space straight H space tan space straight theta space plus space straight h space tan space straight theta
straight H open parentheses 1 space minus space tan space straight theta close parentheses space equals space straight h open parentheses 1 space plus space tan space straight theta close parentheses
straight H space equals space fraction numerator straight h open parentheses 1 space plus space tan space straight theta close parentheses over denominator open parentheses 1 space minus space tan space straight theta close parentheses end fraction
straight H space equals space fraction numerator straight h open parentheses tan space 45 degree space plus space tan space straight theta close parentheses over denominator open parentheses 1 space minus space tan space straight theta space tan space 45 degree close parentheses end fraction
We space know space tan space open parentheses straight A space plus space straight B close parentheses space equals space fraction numerator tan space straight A space plus space tan space straight B over denominator 1 space minus space tan space straight A space tan space straight B end fraction
so space straight H space equals space straight h space tan space open parentheses 45 degree space plus space straight theta close parentheses
So comma space the space correct space option space is space left parenthesis straight a right parenthesis.
end style

Question 5

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

begin mathsize 12px style open parentheses straight a close parentheses space straight h over 2 straight m
open parentheses straight b close parentheses space square root of 3 hm
open parentheses straight c close parentheses space straight h over 3 straight m
open parentheses straight d close parentheses space fraction numerator straight h over denominator square root of 3 end fraction straight m end style

Solution 5

begin mathsize 12px style Let space height space of space tower space be space straight H.
Angle space of space depression space is space 60 degree. space So space by space alternate space interior space angle space property.
angle DBC space equals space 60 degree
tan space 60 degree space equals space DC over BC
BC space equals space fraction numerator DC over denominator tan space 60 degree end fraction
space space space space space space space space equals space fraction numerator straight h over denominator square root of 3 end fraction space space space space space space space space.... open parentheses 1 close parentheses
tan space 30 degree space equals space AB over BC
BC space equals space fraction numerator AB over denominator tan space 30 degree end fraction
BC space equals space square root of 3 AB
BC space equals space square root of 3 straight H space space space space space space space space space space space space space space.... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
square root of 3 straight H space equals space fraction numerator straight h over denominator square root of 3 end fraction
straight H space equals space straight h over 3
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 6

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

begin mathsize 12px style open parentheses straight a close parentheses space 3 square root of 2 straight x
open parentheses straight b close parentheses space 2 square root of 3 straight x
open parentheses straight c close parentheses space fraction numerator square root of 3 over denominator 2 end fraction straight x
open parentheses straight d close parentheses space fraction numerator 2 over denominator square root of 3 end fraction straight x end style

Solution 6

begin mathsize 12px style Let space the space height space be space straight h.
tan space 30 degree space equals space straight h over AC space equals space fraction numerator straight h over denominator AB space plus space straight x end fraction
AB space plus space straight x space equals space straight h square root of 3 space space space space space space space space space.... open parentheses 1 close parentheses
tan space 60 degree space equals space straight h over AB
AB space equals space fraction numerator straight h over denominator square root of 3 end fraction space space space space space space space space space space space space..... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
fraction numerator straight h over denominator square root of 3 end fraction space plus space straight x space equals space straight h square root of 3
straight x space equals space straight h square root of 3 space minus space fraction numerator straight h over denominator square root of 3 end fraction
space space space space space space equals space fraction numerator 2 straight h over denominator square root of 3 end fraction
straight h space equals space fraction numerator square root of 3 over denominator 2 end fraction straight x
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 7

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

begin mathsize 12px style open parentheses straight a close parentheses space open parentheses square root of 3 space plus space 1 close parentheses straight x
open parentheses straight b close parentheses space open parentheses square root of 3 space end root space minus space 1 close parentheses straight x
open parentheses straight c close parentheses space 2 square root of 3 space straight x
open parentheses straight d close parentheses space 3 square root of 2 straight x end style

Solution 7

begin mathsize 12px style Let space height space be space straight h space and space BC space equals space 2 straight x
tan space 30 degree space equals space AD over AC space equals space fraction numerator straight h over denominator AB space plus space BC end fraction
AB space plus space 2 straight x space equals space straight h square root of 3 space space space space space space space space space space...... open parentheses 1 close parentheses
tan space 45 degree space equals straight h over AB
AB space equals space straight h space space space space space space space...... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
straight h space plus space 2 straight x space equals space straight h square root of 3
2 straight x space equals space straight h open parentheses square root of 3 space minus space 1 close parentheses
straight h space equals space fraction numerator 2 straight x over denominator square root of 3 space minus space 1 end fraction cross times space fraction numerator square root of 3 space plus space 1 over denominator square root of 3 space plus space 1 end fraction
space space space space space space equals space fraction numerator 2 open parentheses square root of 3 space plus space 1 close parentheses straight x over denominator 3 space minus space 1 end fraction
straight h space equals space open parentheses square root of 3 space plus space 1 close parentheses straight x
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 8

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

begin mathsize 12px style open parentheses straight a close parentheses space square root of 2 straight a space metres
open parentheses straight b close parentheses space fraction numerator straight a over denominator 2 square root of 2 end fraction space metres
open parentheses straight c close parentheses space fraction numerator straight a over denominator square root of 2 end fraction space metres
open parentheses straight d close parentheses space 2 straight a space metres end style

Solution 8

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.

begin mathsize 12px style So space AC space equals space BC space equals space straight a over 2
tan space straight theta space equals space fraction numerator straight x over denominator open parentheses begin display style straight a over 2 end style close parentheses end fraction equals fraction numerator 2 straight x over denominator straight a end fraction space space space space space space space.... open parentheses 1 close parentheses
tan open parentheses 90 space minus space straight theta close parentheses space equals space fraction numerator 2 straight x over denominator open parentheses begin display style straight a over 2 end style close parentheses end fraction equals space fraction numerator 4 straight x over denominator straight a end fraction space space space space space space space space space space.... open parentheses 2 close parentheses
We space know comma space tan space open parentheses 90 space minus space straight theta close parentheses space equals space cotθ
space space space space space space space space space space space space space space space space space space space space space space tan space open parentheses 90 space minus space straight theta close parentheses space equals space fraction numerator 1 over denominator tan space straight theta end fraction
space space space space space space space space space space space space space space space space space space space space space space tan space straight theta space. space tan space open parentheses 90 space minus space straight theta close parentheses space equals space 1 space space space space space space space space space space space...... open parentheses 3 close parentheses
from space open parentheses 1 close parentheses comma space open parentheses 2 close parentheses comma space open parentheses 3 close parentheses
open parentheses fraction numerator 2 straight x over denominator straight a end fraction close parentheses open parentheses fraction numerator 4 straight x over denominator straight a end fraction close parentheses space equals space 1
fraction numerator 8 straight x squared over denominator straight a squared end fraction equals 1
straight x squared space equals space straight a squared over 8 space space space space space space space space space rightwards double arrow straight x equals fraction numerator straight a over denominator 2 square root of 2 end fraction end style

So, the correct option is (b).

Question 9

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

Solution 9

EC || AB

Hence

EA = CB = 10

AD = AE + ED

ED = AD - AE

     = 16 - 10 = 6

 begin mathsize 12px style In space triangle DEC
sin space 30 degree space equals space DE over straight l
straight l space equals space fraction numerator DE over denominator sin space 30 end fraction space rightwards double arrow space straight l space equals space 2 space cross times space 6 space equals space 12 end style

So, the correct option is (c).

Question 10

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

Solution 10

begin mathsize 12px style AE space rightwards arrow space lamp space post
BD space rightwards arrow space girl
AB space equals space 3
BC space equals space 4.5 space straight m
tan space straight Q space equals space BD over BC space equals space AE over AC
space space space space space space space space space space space space space space space space space fraction numerator 1.5 over denominator 4.5 end fraction equals fraction numerator straight h over denominator AB space plus space BC end fraction
space space space space space space space space space space space space space space space space space space space 1 third equals fraction numerator straight h over denominator 3 space plus space 4.5 end fraction
space space space space space space space space space space space space space space space space space space space space space straight h space equals space fraction numerator 7.5 over denominator 3 end fraction equals 2.5
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Chapter 12 - Some Applications of Trigonometry Excercise 12.43

Question 1

begin mathsize 12px style The space length space of space shadow space of space straight a space tower space on space the space plane space ground space is space square root of 3 space times space the space height space of space the space tower. space The space angle space of space elevation space of space sun space is
open parentheses straight a close parentheses space 45 degree
open parentheses straight b close parentheses space 30 degree
open parentheses straight c close parentheses space 60 degree
open parentheses straight d close parentheses space 90 degree end style

Solution 1

begin mathsize 12px style straight h space rightwards arrow space height space of space tower
straight s space rightwards arrow space shadow space of space tower
straight theta space rightwards arrow angle space of space elevation
Given colon space straight s space equals space square root of 3 straight h space rightwards double arrow straight h over straight s equals space fraction numerator 1 over denominator square root of 3 end fraction space space space space space space space space space space space..... open parentheses 1 close parentheses
In space the space given space triangle comma
tan space straight theta space equals space straight h over straight s
from space open parentheses 1 close parentheses
tan space straight theta space equals space fraction numerator 1 over denominator square root of 3 end fraction
straight theta space equals space 30 degree
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 2

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

begin mathsize 12px style open parentheses straight a close parentheses space 25 square root of 3
open parentheses straight b close parentheses space 50 square root of 3
open parentheses straight c close parentheses space 75 square root of 3
open parentheses straight d close parentheses space 150 end style

Solution 2

begin mathsize 12px style angle straight C space equals space 30 degree space by space alternate space Interior space angle space property
tan space 30 degree space equals space AB over BC
BC space equals space fraction numerator AB over denominator tan space 30 end fraction space equals fraction numerator AB over denominator begin display style fraction numerator 1 over denominator square root of 3 end fraction end style end fraction equals square root of 3 space AB equals 75 square root of 3 space straight m
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 3

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

begin mathsize 12px style open parentheses straight a close parentheses space 15 square root of 3 space straight m
open parentheses straight b close parentheses space fraction numerator 15 square root of 3 over denominator 2 end fraction straight m
open parentheses straight c close parentheses space 15 over 2 straight m
open parentheses straight d close parentheses space 15 space straight m end style

Solution 3

begin mathsize 12px style AB space rightwards arrow space wall
AC space rightwards arrow space ladder
angle straight A space plus space angle straight B space plus space angle straight C space equals space 180 degree
angle straight C space equals space 180 degree space minus space 90 degree space minus space 60 degree
space space space space space space space space equals space 30 degree
sin space 30 degree space equals space AB over AC
AB space equals space AC space sin space 30 degree
space space space space space space space equals space AC over 2
AB space equals space 15 over 2 straight m
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 4

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

begin mathsize 12px style open parentheses straight a close parentheses space 50 square root of 3
open parentheses straight b close parentheses space 150 square root of 3
open parentheses straight c close parentheses space 150 square root of 2
open parentheses straight d close parentheses space 75 end style

Solution 4

begin mathsize 12px style angle straight C space equals space 30 degree
Hence comma
tan space 30 degree space equals space AB over BC rightwards double arrow square root of 3 space equals space AB over BC
rightwards double arrow BC space equals space fraction numerator 1 over denominator square root of 3 end fraction space AB
space space space space space space space space space space space equals space fraction numerator 1 over denominator square root of 3 end fraction space left parenthesis 150 right parenthesis equals 50 square root of 3
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 5

begin mathsize 12px style If space the space height space of space straight a space vertical space pole space is space square root of 3 space times space the space length space of space its space shadow space on space the space ground comma space then space the space angle space of space elevation space of space the space sun space at space that space time space is
open parentheses straight a close parentheses space 30 degree
open parentheses straight b close parentheses space 60 degree
open parentheses straight c close parentheses space 45 degree
open parentheses straight d close parentheses space 75 degree end style

Solution 5

begin mathsize 12px style straight h space rightwards arrow space height space of space pole space
straight b space rightwards arrow space length space of space shadow
straight theta space rightwards arrow space angle space of space elevation
given space straight h space equals space square root of 3 space straight b
straight h over straight b space equals space square root of 3
tan space straight theta space equals straight h over straight b
tan space straight theta space equals space square root of 3
straight theta space equals space 60 degree
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 6

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

begin mathsize 12px style open parentheses straight a close parentheses space 50 square root of 3
open parentheses straight b close parentheses space 50
open parentheses straight c close parentheses space fraction numerator 50 over denominator square root of 2 end fraction
open parentheses straight d close parentheses space fraction numerator 50 over denominator square root of 3 end fraction end style

Solution 6

begin mathsize 12px style tan space 45 degree space equals space straight h over 50
rightwards double arrow 1 space equals space straight h over 50
rightwards double arrow straight h space equals space 50
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 7

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

begin mathsize 12px style open parentheses straight a close parentheses space fraction numerator 4 over denominator square root of 3 end fraction
open parentheses straight b close parentheses space 4 square root of 3
open parentheses straight c close parentheses space 2 square root of 2
open parentheses straight d close parentheses space 4 end style

Solution 7

begin mathsize 12px style Length space of space ladder space equals space BC
cos space 60 degree space equals space AC over BC equals 2 over BC
BC space equals space fraction numerator 2 over denominator cos space 60 degree end fraction space equals fraction numerator 2 over denominator begin display style 1 half end style end fraction equals space 4
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style