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 ,
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 ,
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
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.