Lesson: Some Applications of Trigonometry

Exercise 9.1 (16)

Question: 1

A circus artist is climbing a 20 m long rope, which is

tightly stretched and tied from the top of a vertical

pole to the ground. Find the height of the pole, if

the angle made by the rope with the ground level is

.(see Fig. 9.11).

30

Fig. Exc_9.1_1

Solution:

From the given figure,

AB is the pole.

In ,

ABC

AB

sin30

AC

AB 1

20 2

20

AB

2

AB 10

So, height of the pole is .

10m

Question: 2

A tree breaks due to storm and the broken part

bends so that the top of the tree touches the ground

making an angle with it. The distance between

30

the foot of the tree to the point where the top

touches the ground is 8 m. Find the height of the

tree.

Solution:

Fig. Exc_9.1_2

From the above figure,

Let the original tree be AC and due to storm, it got

broken down into two pieces. The broken part is

taken as and it makes an angle with the

A C

30

ground.

In ,

A BC

BC

tan30

A C

BC 1

8

3

8

BC m

3

Now,

A C

cos30

A B

8 3

A B 2

16

A B m

3

Height of the tree

A B BC

16 8

3 3

24

3

8 3 m

Hence, the height of the tree is m.

8 3

Question: 3

A contractor plans to install two slides for the

children to play in a park. For the children below

the age of 5 years, she prefers to have a slide whose

top is at a height of 1.5 m, and is inclined at an

angle of to the ground, whereas for the elder

30

children she wants to have a steep side at a height

of 3 m, and inclined at an angle of to the

60

ground. What should be the length of the slide in

each case?

Solution:

Fig. Exc_9.1_3 (a)

Fig. Exc_9.1_3(b)

Let AC and PR be the slides for younger and elder

children in the above pictures respectively.

In ,

ABC

AB

sin30

AC

1.5 1

AC 2

AC 3m

Now, In ,

PQR

PQ

sin60

PR

3 3

PR 2

6

PR

3

PR 2 3

Hence, the lengths of these slides are 3 m and

2 3

m.

Question: 4

The angle of elevation of the top of a tower from a

point on the ground, which is 30 m away from the

foot of the tower is . Find the height of the

30

tower.

Solution:

Fig. Exc_9.1_4

Let us consider, AB be the tower and the angle of

elevation from the point C (on ground) is .

30

In ,

ABC

AB

tan30

BC

AB 1

30

3

30

AB

3

AB 10 3 m

Hence, the height of the tower is m.

10 3

Question: 5

A kite is flying at a height of 60 m above the

ground. The string attached to the kite is

temporarily tied to a point on the ground. The

inclination of the string with the ground is .

60

Find the length of the string, assuming that there is

no slack in the string.

Solution:

Fig. Exc_9.1_5

Let us consider, K as the kite and the string of the

kite is tied to point P on the ground.

In ,

KLP

KL

sin60

KP

60 3

KP 2

120

KP

3

KP 40 3 m

Hence, the length of the string is m.

40 3

Question: 6

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 to

30

as he walks towards the building. Find the

60

distance he walked towards the building.

Solution:

Fig. Exc_9.1_6

Let the boy is standing at a point S initially. He

walked towards the building and reached at point

T.

Now,

PR PQ RQ

PR

30 1.5

28.5m

57

m

2

In ,

PAR

PR

tan30

AR

57

1

2

AR

3

57 1

2AR

3

57

AR 3 m

2

Now, in ,

PRB

PR

tan60

BR

57

2

3

BR

57

3

2BR

57

BR

2 3

19 3

BR m

2

Now,

ST AB

AR BR

57 3 19 3

2 2

38 3

2

19 3 m

Hence, the boy walked m towards the

19 3

building.

Question: 7

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 and

45

respectively. Find the height of the tower.

60

Solution:

Fig. Exc_9.1_7

Let us consider, BC as the building, AB as the

transmission tower, and D as the point on the

ground from where the elevation angles are to be

measured.

In ,

BCD

BC

tan45

CD

20

1

CD

CD 20m

Now, In ,

ACD

AC

tan60

CD

AB BC

3

CD

AB 20

3

20

AB 20 3 20

AB 20 3 1 m

Hence, the height of the transmission tower is

.

20 3 1 m

Question: 8

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 and from the same point

60

the angle of elevation of the top of the pedestal is

. Find the height of the pedestal.

45

Solution:

Fig. Exc_9.1_8

Let us consider, AB as the statue, BC as the

pedestal, and D as the point on the ground from

where the elevation angles are to be measured.

In ,

BCD

BC

tan45

CD

BC

1

CD

BC CD

Now, In ,

ACD

AB BC

tan60

CD

AB BC

3

CD

1.6 BC BC 3 BC CD

BC 3 1 1.6

1.6

BC

3 1

Multiply numerator and denominator by .

3 1

1.6 3 1

BC

3 1 3 1

2

2

1.6 3 1

3 1

1.6 3 1

BC

2

0.8 3 1

Hence, the height of the pedestal is m.

0.8 3 1

Question: 9

The angle of elevation of the top of a building from

the foot of the tower is and the angle of

30

elevation of the top of the tower from the foot of

the building is . If the tower is 50 m high, find

60

the height of the building.

Solution:

Fig. Exc_9.1_9

Let us consider, AB as the building and CD as the

tower.

In ,

CDB

CD

tan60

BD

50

3

BD

50

BD

3

Now, in ,

ABD

AB

tan30

BD

AB 1

50

3

3

50 1

AB

3 3

50

AB

3

2

AB 16

3

Hence, the height of the building is m.

2

16

3

Question: 10

Two poles of equal heights are standing opposite

each other and 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

and , respectively. Find the height of poles

60

30

and the distance of the point from the poles.

Solution:

Fig. Exc_9.1_10

Let us consider, AB and CD as the two poles and O

be the point from where the elevation angles are

measured.

In ,

ABO

AB

tan60

BO

AB

3

BO

AB

BO

3

Now, in ,

CDO

CD

tan30

DO

CD 1

80 BO

3

CD 3 80 BO

AB

CD 3 80

3

AB

CD 3 80 ......(1)

3

The poles AB and CD are of equal heights.

So, CD

AB

Put this in equation .

1

CD

CD 3 80

3

1

CD 3 80

3

3 1

CD 80

3

4CD 80 3

CD 20 3

Now,

AB

BO

3

CD

3

20 3

3

BO 20m

DO BD BO

80 20

60m

Hence, the height of both the poles is and

20 3 m

the point O is

and far from these poles.

20m

60m

Question: 11

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 . From another point 20 m away from

60

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 (see Fig. 9.12). Find the height of the

30

tower and the width of the canal.

Fig. Exc_9.1_11

Solution:

According to the given figure,

In ,

ABC

AB

tan60

BC

AB

BC

3

AB

3

BC

Now, in ,

ABD

AB

tan30

BD

AB 1

BC CD

3

AB 1

AB

3

20

3

AB 3 1

AB 20 3 3

3AB AB 20 3

2AB 20 3

AB 10 3

Now,

AB

BC

3

10 3

3

10m

Hence, the height of the tower is m and the

10 3

width of the canal is 10 m.

Question: 12

From the top of a high building, the angle of

7m

elevation of the top of a cable tower is and the

60

angle of depression of its foot is . Determine the

45

height of the tower.

Solution:

Fig. Exc_9.1_12

Let us consider, AB as a building and CD as a cable

tower.

In ,

ABD

AB

tan45

BD

7

1

BD

BD 7m

Now, In ,

ACE

AE BD 7m

CE

tan60

AE

CE

3

7

CE 7 3 m

CD CE ED

7 3 7

7 3 1 m

Therefore, the height of the cable tower is

.

7 3 1 m

Question: 13

As observed from the top of a high lighthouse

75m

from the sea-level, the angles of depression of two

ships are and . If one ship is exactly behind

30

45

the other on the same side of the lighthouse, find

the distance between the two ships.

Solution:

Fig. Exc_9.1_13

Let us consider, AB as the lighthouse and the two

ships be at point C and D respectively.

In ,

ABC

AB

tan45

BC

75

1

BC

BC 75m

Now In ,

ABD

AB

tan30

BD

75 1

BC CD

3

75 1

75 CD

3

75 3 75 CD

75 3 75 CD

CD 75 3 1

Hence, the distance between the two ships is

m.

75 3 1

Question: 14

A 1.2 m tall girl spots a balloon moving with the

wind in a horizontal line at a height of 88.2 m from

the ground. The angle of elevation of the balloon

from the eyes of the girl at any instant is . After

60

some time, the angle of elevation reduces to

30

(see Fig. 9.13). Find the distance travelled by the

balloon during the interval.

Fig. Exc_9.1_14 (Ques.)

Solution:

Fig. Exc_9.1_14 (Sol.)

Let the initial position of balloon be A and after

some time, its position be B and CD be the girl who

spots the balloon.

In ,

ACE

AE

tan60

CE

AF EF

tan60

CE

88.2 1.2

3

CE

87

3

CE

87

CE

3

CE 29 3 m

Now In ,

BCG

BG

tan30

CG

88.2 1.2 1

CG

3

CG 87 3

Distance travelled by balloon

EG

CG CE

87 3 29 3

58 3 m

Thus, the distance is .

58 3 m

Question: 15

A straight highway leads to the foot of a tower. A

man standing at the top of the tower observes a car

as an angle of depression of , which is

30

approaching the foot of the tower with a uniform

speed. Six seconds later, the angle of depression of

the car is found to be . Find the time taken by

60

the car to reach the foot of the tower from this

point.

Solution:

Fig. Exc_9.1_15

Let us consider, AB as the tower and initial position

of the car is C, which changes to D after six

seconds.

In

ADB,

AB

tan60

DB

AB

3

DB

AB

DB

3

Now in ,

ABC

AB

tan30

BC

AB 1

BD DC

3

AB 3 BD DC

AB

AB 3 DC

3

AB

DC AB 3

3

1

AB 3

3

2AB

3

Time taken by the car to travel distance

.

DC 6seconds

[DC ]

2AB

3

So, speed of car is given by,

Distance

Speed

Time

2AB

3

6

2AB

6 3

The speed remains the same for the distance DB.

Now, time taken by the car to travel distance DB is

given by,

Distance

Time

Speed

AB

AB

3

DB

2AB

3

6 3

AB 6 3

2AB 3

6

2

3seconds

Thus, the time taken by the car is seconds.

3

Question: 16

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:

Fig. Exc_9.1_16

Let us consider, AQ as the tower and the points R

and S be 4 m and 9 m away from the base of the

tower respectively.

According to the question, the angles are

complementary.

So, if one angle is , the other will be .

θ

90 θ

In ,

AQR

AQ

tanθ

QR

AQ

tanθ ......(1)

4

Now in ,

AQS

AQ

tan 90 θ

SQ

AQ

cot θ ......(2)

9

Multiply equations and .

1

2

AQ AQ

tanθ cot θ

4 9

2

AQ

1

36

2

AQ 36

AQ 36

AQ 6

However, height cannot be negative. Hence, the

height of the tower is 6 m.