Some Applications of Trigonometry Class 10

Ex 9.1, 1 teachoo.com
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 30°
Let, the height of pole be AB. A
20m

Assuming that rope is tied at point C. ?
It is given that,

aN
Length of rope = 20m B Cc
Hence, AC = 20m
Also, angle made by rope with the ground level = 30°

Hence, ZACB=30°

We need to find height of pole i.e. AB

Ex9.1,2 teachoo.com
A tree breaks due to storm and the broken part bends so that the
top of the tree touches the ground making an angle 30° with it.
The distance between the foot of the tree to the point where the
top touches the ground is 8 m. Find the height of the tree.
A
Let, the broken part of tree be AC @ 4
Lee
It is given that, eae cate eae eae
Distance between foot of the tree B and point C = 8m
So, BC = 8m
Also, broken parts of tree makes an angle 30° with ground
So, ZC = 30°
We need to find height of tree
Height of tree = Height of broken part + height of remaining tree
Height of tree = AB + AC

Ex9.1,3 teachoo.com
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 30° to the ground, whereas for elder children, she wants to have
a steep slide at a height of 3m, and inclined at an angle of 60° to
the ground. What should be the length of the slide in each case?

Pp
Let smaller slide be represented

A
by right angle triangle ABC. 3m >
2 Slide 2 \‘
15™| slide 1

Here, Height of small slide = 1.5m gO 30 ¢ A 607\.

Q R
So, AB= 1.5m
Also it is inclined at an angle of 30° to the ground.
Hence, ZACB = 30°
We need to find the length of slide i.e. AC.

Ex9.1,4 teachoo.com
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 30°. Find
the height of the tower. A
Let tower be AB ?
Let point be C

LN‘ Cj

c 30m 8
Distance of point C from foot of tower = 30m
Hence, BC = 30m
Angle of elevation = 30°
So, ZACB = 30°
Since tower is vertical,
2 ABC = 90°

Ex9.1,5 teachoo.com
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. A
te
Given that,
. wo Lien te tha ? 60m
Height at which kite is flying = 60 metre
Hence, AB = 60m
/\ Cj
Cc B
Also, inclination of the string with the ground = 60°
Hence, ZACB = 60°
We have to find length of string, i.e., AC
Here, AB is perpendicular to ground
So, Z ABC = 90°

Ex 9.1,6 teachoo.com
A1.5 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 30° to 60° as he walks towards the building.
Find the distance he walked towards the building. A
Boy is 1.5 m tall 30m
So, PQ=1.5m i

Pag) /\ J

1.5 mgs 1:

Building is 30m tall, , ‘ %

re) ? R B
So, AB = 30m .
Given that,
angle of elevation from initial point (Q) to top of building = 30°
Hence, ZAPC = 30°

Ex9.1,7 teachoo.com
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 45° and 60° respectively. Find the height of the tower.
f
Let the building be AB and tower be CA. ae ?
Ni
Building is 20 m high, Al |
wiki
on
So, AB = 20m ti
nade
i
tea
Jase FREE
Let point of ground be P. JA moat
P B
Angle of elevation from point P to bottom of tower A = 45°
Hence, ZAPB = 45°
Also, Angle of elevation from point P to top of the tower C = 60°
Hence, ZCPB = 60°

Ex9.1,8 teachoo.com
A statue, 1.6 m tall, stands on the top of a pedestal. From a point on
the ground, the angle of elevation of the top of the statue is 60° and
from the same point the angle of elevation of the top of the pedestal
is 45°. Find the height of the pedestal.
oe
Here pedestal is AB & Statue is AC 7
Height of pedestal = AB iS
1.6
13
Length of the statue = 1.6m :
Hence, AC = 1.6m a=:
on
op
Angle of elevation to top of statue = 60° Lon ii cued
Hence, CPB = 60° P 8
Angle of elevation at the top of the pedestal = 45°
Hence, ZAPB = 45°

Ex9.1,9 teachoo.com
The angle of elevation of the top of a building from the foot of the
tower is 30° and the angle of elevation of the top of the tower from
the foot of the building is 60°. If the tower is 50 m high, find the
height of the building. D
Let building be AB & tower be CD
0

Given A ~
Height of the tower =50m ? I
Hence, CD = 50m \ Kory 30°/¢@ I =

B c
Angle of elevation of top of building from foot of tower = 30°
Hence, ZACB = 30°
Angle of elevation of top of tower from foot of building = 60°
Hence, ZDBC = 60°

Ex 9.1, 10 teachoo.com
Two poles of equal heights are standing opposite each other on
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
60° and 30°, respectively. Find the height of the poles and the
distances of the point from the poles.

A D

IN [|
Let two poles be AB & CD
So, Length of pole = AB = CD
{iN

B P Cc
Also, Length of the road = 80m . 80m >
So, BC = 80m
Lets point P be a point on the road between poles,
We need to find height of poles i.e. AB & CD
and distance of the point from poles, i.e. BP & CP

Ex9.1,11 teachoo.com
ATV 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 60°. 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° (see figure). Find the
height of the tower and the width of the canal.

A
Let tower be AB
So, height of tower = AB

a

<—20m——> € B

Given that,from a point on the other bank directly opposite the
Tower,the angle of elevation of the tower is 60°.
Hence, ZACB = 30°
Also,
Angle of elevation from point D to top of the tower = 30°
So, Z ADB = 30°

Ex 9.1, 12 teachoo.com
From the top of a 7 m high building, the angle of elevation of the top
of a cable tower is 60° and the angle of depression of its foot is 45°.
Determine the height of the tower. E
Let building be AB & tower be CE
Given height of building = AB = 7m A ee ?
" ~ y
I
to n
° c
From the top of building, angle of elevation of top of tower = 60°.
Hence, ZEAD = 60°
Angle of depression of the foot of the tower = 45°
Hence, ZCAD = 45°
We need to find height of tower i.e. CE

Ex 9.1, 13 teachoo.com
As observed from the top of a 75 m high lighthouse from the sea-
level, the angles of depression of two ships are 30° and 45°. If one
ship is exactly behind the other on the same side of the lighthouse,
find the distance between the two ships. p
----------4-gA
45°
30'

Given that height of the lighthouse is 75 m

75m
Hence, AD = 75m

? Cc D
+

And angle of depression of first ship is 45°
So, Z PAC = 45 °
And angle of depression of second ship is 30°
So, Z PAB = 30°
We need to find distance between the two ships, i.e. BC

Ex 9.1, 14 teachoo.com
A 1.2 mtall 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 60°. After some
time, the angle of elevation reduces to 30° (see figure ). Find the
distance travelled by the balloon during the interval.
A. 30° 'B :
ee
F

Given that 1.2 m tall girl sees a balloon
So, AG =1.2m
Also, AG & BF are parallel
BF=AG=1.2m

Ex9.1,15 teachoo.com
A straight highway leads to the foot of a tower. A man standing at the
top of the tower observes a car at an angle of depression of 30°,
which is approaching the foot of the tower with a uniform speed. Six
seconds later, the angle of depression of the car is found to be 60°.
Find the time taken by the car to reach the foot of the tower from
this point. p A
60°
30°

Let the man & tower height be AD : |

B c+——>D
Given that man sees car first at an angle of depression of 30°
So, Z PAB = 30°
After 6 seconds, the man sees car at an angle of depression of 60°
So, Z PAC = 60°

Ex 9.1, 16 teackoo
The angles of elevation of the top of a tower from two points at a
distance of 4 mand 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. A
Given that AB is the tower
P, Qare the point at distance 4 m and 9 m resp.
Also, PB=4m,QB=9m
Q Pe +B
4m

& — Angle of elevation from P is a <——onm

Angle of elevation from Qis f.
Given a and f are complementary.

a+fB=90°

We need to prove AB=6m

Example 1 teachoo.com
A tower stands vertically on the ground. From a point on the ground,
which is 15 m away from the foot of the tower, the angle of elevation
of the top of the tower is found to be 60°. Find the height of the tower]
A
Let, the height of the tower be H meter.
So, AB = H metre
Height = H
Distance of the point
from the foot of the tower = 15m
Hence, CB = 15m /
c 15m B
Angle of elevation = 60°
ZACB = 60°
Since tower is vertical to ground,
So, Z ABC = 90°

Example 2 teachoo.com
An electrician has to repair an electric fault on a pole of height 5 m.
She needs to reach a point 1.3m below the top of the pole to
undertake the repair work. What should be the length of the ladder
that she should use which, when inclined at an angle of 60° to the
horizontal, would enable her to reach the required position? Also,
how far from the foot of the pole should she place the foot of the
ladder? (You may take V3 = 1.73) ,
Given, the height of the pole = 5 meter. 8
So, AD=5m
5m
_
BD = AD— AB
BD=5-13=3.7m SB
D ? Cc
Here, we have to find the length of ladder, i.e. BC
& distance from foot of the ladder to the foot of the pole, i.e. DC

Example 3
An observer 1.5 m tall is 28.5 m away from a chimney. The angle of
elevation of the top of the chimney from her eyes is 45°. What is the
height of the chimney?
Here, In diagram AB is chimney and CD is observer.

A
Angle of elevation = 45° .
Hence, ZADE = 45° :

f ?
And, Distance (BC) = 28.5m '
Height of observer = CD =1.5m wa -
Since BC & DE are parallel lines 28.5m B
BC =DE=28.5m
Also, CD & BE are parallel lines
& CD=BE=1.5m

Example 4 teachoo.com
From a point P on the ground the angle of elevation of the top of a
10 m tall building is 30°. A flag is hoisted at the top of the building
and the angle of elevation of the top of the flagstaff from P is 45°.
Find the length of the flagstaff and the distance of the building from
the point P. (You may take V3 = 1.732)?
Given Height of building = 10 metre.
So, BA = 10m ee
a oo
Vs fa |)
£{ ——
ps-—— 7 ——>A
Angle of elevation from point P to the top of the building = 30°
So, Z BPA = 30°
Angle of elevation from point P to the top of flag = 45°
So, Z DPA = 45°

Exampl e5 teachoo.com
The shadow of a tower standing on a level ground is found to be
40 m longer when the Sun’s altitude is 30° than when it is 60°. Find
the height of the tower. A
Given tower be AB
When Sun’s altitude is 60°
Z ACB = 60° “A /\
D

Cc B
& Length of shadow = BC — 40m —
When Sun’s altitude is 30°
Z ADB = 30°
& Length of shadow = DB
Shadow is 40 m when angle changes from 60° to 30°
CD =40m

Example 6 teachoo.com
The angles of depression of the top and the bottom of an 8 m tall
building from the top of a multi-storeyed building are 30° and 45°,
respectively. Find the height of the multi-storeyed building and the
distance between the two buildings.
QL
39
Height of tall building (AB} = 8m
8
A ? ;
Let, Height of the multi-storeyed building = PC
And, distance between two building = AC
Angle of depression to top of building = 2 QPB = 30°
Angle of depression to bottom of building = 2 QPA = 45°

Exampl e7 teachoo.com
From a point on a bridge across a river, the angles of depression of
the banks on opposite sides of the river are 30° and 45°,
respectively. If the bridge is at a height of 3 m from the banks, find
the width of the river.
Q id __R
Xe
TN
A D B
?
Let the width of the river = AB
And the bridge is at a height of 3m from the banks,
So, DP = 3 metre
Angle of depression of the banks on opposite sides of the river are
30° and 45° respectively.
So, Z QPA = 30° and 2 RPB=45°

A group of students of class X visited India Gate on an education trip. The teacher and students had interest in history as well. The teacher narrated that India Gate, official name Delhi Memorial, originally called All-India War Memorial, monumental sandstone arch in New Delhi, dedicated to the troops of British India who died in wars fought between 1914 and 1919.The teacher also said that India Gate, which is located at the eastern end of the Rajpath (formerly called the Kingsway), is about 138 feet (42 metres) in height.
Question 1
What is the angle of elevation if they are standing at a distance of 42m away from the monument?
(a) 30°
(b) 45°
(c) 60°
(d) 0°
Question 2
What is the angle of elevation if they are standing at a distance of
42m away from the monument?
(a) 25.24 m
(b) 20.12 m
(c) 42 m
(d) 24.64 m
Question 3
If the altitude of the Sun is at 60 , then the height of the vertical tower that will cast a shadow of length 20 m is
(a) 20√3 m
(b) 20/√3 m
(c) 15 /√3 m
(d) 15√3 m
Question 4
The ratio of the length of a rod and its shadow is 1:1 . The angle of elevation of the Sun is
(a) 30°
(b) 45°
(c) 60°
(d) 90°
Question 5
The angle formed by the line of sight with the horizontal when the object viewed is below the horizontal level is
(a) corresponding angle
(b) angle of elevation
(c) angle of depression
(d) complete angle

A Satellite flying at height h is watching the top of the two tallest mountains in Uttarakhand and Karnataka ,them being Nanda Devi(height 7,816m) and Mullayanagiri (height 1,930 m). The angles of depression from the satellite, to the top of Nanda Devi and Mullayanagiri are 30° and 60° respectively. If the distance between the peaks of two mountains is 1937 km , and the satellite is vertically above the midpoint of the distance between the two mountains.
Question 1
The distance of the satellite from the top of Nanda Devi is
(a) 1139.4 km
(b) 577.52 km
(c) 1937 km
(d) 1025.36 km
Question 2
The distance of the satellite from the top of Mullayanagiri is
(a) 1139.4 km
(b) 577.52 km
(c) 1937 km
(d) 1025.36 km
Question 3
The distance of the satellite from the ground is
(a) 1139.4 km
(b) 577.52 km
(c) 1937 km
(d) 1025.36 km
Question 4
What is the angle of elevation if a man is standing at a distance of 7816m from Nanda Devi?
(a) 30°
(b) 45°
(c) 60°
(d) 0°
Question 5
If a mile stone very far away from, makes 45 to the top of Mullanyangiri montain. So, find the distance of this mile stone form the mountain.
(a) 1118.327 km
(b) 566.976 km
(c) 1937 km
(d) 1025.36 km