Exercise 20

An electric pole is 10 metres high. If its shadow is 10 N3 metres in length, find the
elevation of the sun.
The angle of elevation of the top of a tower, from a point on the ground and at a
2. Le
distance of 150 mfrom its foot, is 30°. Findthe height of the tower correct to one place
of decimal.
. Aladder is The foot
placed against a wall such that it just reaches the top of the wall.
of the ladder is 1:5 metres away from the wall and the ladder is inclined at an angle
of 60° with
the ground. Find the height oof the wall.
Heights and Distances 489
 4. What is the angle of elevation of the sun when the length of shadow of a vertical pole
is equal to its height?
5. From a point P on top of a tower is 30°. If
level the angle of elevation of the
ground,
ower 1s 100 m high, how far is P from the foot of the tower ? Calculate
Fromn the top of a cliff 92 mhigh, the angle of depression of a buoy is 20°.
the CI:
Cate iedrest metre, the distance of the bu1OV from the foot of
m. If the string is tight and the angle
y1S tlying a kite with a string of length 100 the kite correct to one decimal place
of
.vaton of the kite is 26° 32 find the height
(ignore the height of the boy).
affixed at a
CIectric pole is 10 m high. A steel wire tied to the top of the pole 1S
V wire makes an angle of 45° with
POnt on the ground to keen the pole upright. If the the wire.
the horizontal through the foot ofthe pole. find the length of
at some distance from the tower knows
Avertical tower is 20 m high. Aman standing
top of the tower is 0:53. How far is he
hat the cosine of the angle of elevation of the
sanding from the foot of the tower?
Hint
If 6is the angle of elevation, then cos 0= 0-53 = 58.

without being detached.

0 The upper part of a tree broken by wind, falls to the ground
38° 30 at a point 6 m
The top of the broken part touches the ground at an angle of
from the foot of the tree. Calculate:
i) the height at which the tree is broken.
(ii) the original height of the tree correct to two decimal places.
N. An observer 1-5m tall is 20-5 metres away from atower 22 metres high. Determine the
angle of elevation of the top of the tower from the eve of the observer. R

1z. In the adjoining figure, the angle of elevation from a point

Pof the top of atower QR, 50 m high is 60 and that of
the tower PT from a point Q is 30°. Find the height of
T
the tower PT, correct to the nearest metre. (2018)

60°
P

13. From a point P on the ground, the angle of elevation of the top of a 10 m
and a helicopter, hovering over the top of the building, are tall building
Find the height of the helicopter above the ground. 30° and 60 respectively.
14. An aeroplane when flying at a height of 3125 m
another plane at an instant when the angles offrom the ground passes vertically below
same point on the ground are 30° and 60° elevation of the two planes from the
two planes at the instant. respectively. Find the distance between the
15. A man observes the angle of
it in ahorizontal line elevation of the top of a tower to be 45°, He
through its base. On covering 20 m, the walks towards
to 60°. Find the angle of
height of the tower correct to 2 elevation change
16. The shadow of a vertical
tower on a level ground significant figures. (2019)
of the sun changes from 45° increases by 10 m
to 30°. Find the the altitude
places. height of the tower, correctwhen
to two decimal
UNDERSTANDING ICs
 ofa hill, the angles
of
the topfound depression
of two
From
eastare
to be 30° and 45
respectively.
Find the consecutive kilometre stones,
due
footof
the hill. distance
of two stones from
the
16.A man observes the angle of elevation of the top of a building to be 30°. He walks
it, in a horizontal line through its base.
towards On
changesto60° Find the heightof the building covering 60 m, the angle of elevation
correct to the nearest metre.
MAta pointonlevel ground, the angle of elevation of a vertical tower is found to be
5
such that its
tangent is 12
On walking 192 m towards the tower, the
3 tangent of the
found to be Find the height of the tower.
angleis 4
Inthetigure, not drawn to scale, TF is atower. The elevation
2
x° where tan x =
Tfrom Ais 5
and AF = 200 m. The
elevation of T from B, where AB = 80 m, is y°. Calculate:
() the height of the tower TE. A B

(i) the angle y, correct to the nearest degree.

M
n the adjoining figure, not drawn to the scale, AB is a tower 45

and two objects Cand Dare located on the ground, on the

same side of AB. When observed from the top Aof the tower, 300 m

their angles of depression are 45° and 60°. Find the distance
between the two objects, if the height of the tower is 300 m.
Give your answer to the nearest metre.
D B

22. The horizontal distance between two towers is 140 m. The angle of elevation of the top
of the first tower, when seen from the top of the second tower is 30°. If the height of
the second tower is 60 m, find the height of the first tower.
. As observed from the top of a 80 m tall light house, the angles of depression of two
with its base are 30° and
same side of the light house in horizontal line
ships on the
Give your answer correct to
40° respectively. Find the distance between the two ships. (2012)
the nearest metre.
A on the ground is 45° and from a point
* lhe angle of elevation of a pillar from a point height
diametrically opposite to A and on the other side of the pillar is 60°. Find the
b
distance between A and B is 15 m.
O the pillar, given that the elevation of
points A andB on the same side of a building, the angles of
o rom two
building are 30° and 60° respectively. If the height of the building is
the top of the and B correct to two decimal
places.
find the distance between A
1O m, the top of alight house
ships A and Bas observed fromthe opposite sides of the
Ve angles of depression of two
respectively. If the two ships are on answer correct to the
wn high are 60° and 45 between the twO ships. Give your
light house, find the distance (2017)
hearest whole number. on
250 m observes the angle of depression of two boats river.
An at an altitude of Find the width of the
the aeroplane
opposite banks of a river
to be 45° and 60° respectively.
nearest whole number.
(2014)
Write the answer correct to the of depression of two rocks which are in a horizontal
X0 angles between the
Fromatower 126 mhigh, the tower are 16° and 12° 20'. Find the distance
line through the base of the
TOcks if they are on the tower.
(ii) the opposite sides of
the same side of the tower
Hoights and nistances 401
 29. A man 1-8 mhigh stands at a distance of 3-6 m from a lamnp post and
casts a
of 5-4 m on the ground. Find the height of the lamp post.
\3 From the top of a cliff, the angle of
shadow
depression of the top and bottom of a tower av
observed to be 45° and 60° respectively. If the height of the tower is 20 m. Find:
(i) the height of the cliff.
(ii) the distance between the cliff and the tower.
31. A pole of height 5 m is fixed on the top of a tower. The (2020)
angle of elevation of tho
top ofthe pole as observed fromn a point A on the ground is 60° and the
angle of
depression of the point A from the top of the tower is 45°. Find the height of the tower
(Take 3 = 1-732)
32. A vertical pole and a vertical tower are on the same level ground. From the top of
the
pole, the angle of elevation of the top of the tower is 60° and the angle of depression
of the foot of the tower is 30°. Find the height of the tower if the height of the pole is
20 m.
(3From the top of a building 20 m high, the angle of elevation of the top of a monument
is 45° and the angle of depression of its foot is 15°. Find the height of the monument.
34. In the adjoining figure, the shadow of a vertical tower on
the level ground increases by 10m, when the altitude of
the sun changes from 45° to 30°. Find the height of the
1
tower and give your answer, correct to of a metre. 30° 45
10
10 m’
(2002)
(35.)An aircraft is flying at a constant height with a speed of 360 km/h. From a point on
the ground, the angle of elevation of the aircraft at an instant was observed to be 45°.
After 20 seconds, the angle of elevation was observed to be 30°. Determine the height
at which the aircraft is flying (use V3 = 1-732).

Multinlo Choioo uostions

 1. () 2(/5-D
V5 () 3;31
120
2. ()
3+ V3
(i)2 3. 11

12. () 30 (ii) 60° (iii) 45° (iv) 60°

Exercise 19

1. (i) -5789 (ii) .9484 (ii) .9087 (iv) 4056

2. (i) -4625 (i) .9984 (iii) -0581 (iv) -6951
3. (i) ·2685 (ii) 1:3383 (ii) 7:3962 (iv) ·1078
4. (i) 35° 22 (ii) 71° 31' (iii) 13° 38' (iv) 39° 34'
5. (i) 62° 27" (ii) 7° 51' (iii) 45° 58 (iv) 70° 3'
6. (i) 15° 2 (ii) 60° 11' (iiil) 72° 31 (iv) 43° 4'
7. 10° 30' 8. (i) -3280 (ii) -0.4483
9. (i) -7423 (ii) 1-6448 10. (i) 43° 28 (ii) 1-2079

Exercise 20
1. 30° 2. 86-6 m 3. 2-6 m 4. 45° 5. 173-2 m 6. 253 m
7. 44-7 m 8. 14-14 m 9. 12-5 m 10. (i) 4-77 m (ii) 12-44 m
11. 45o 12. 17 m 13. 30 m 14. 6250 m 15. 47 m
16. 13-66 m 17. 1-366 km, 2-366 km 18. 52m 19. 180 m
20. (i) 80m (ii) 34° 21. 127 m 22. 140-83 m 23. 43 m
24. 9-51 m 25. 11-55 m 26. 95 m
27. 394 m or 106 m according as the boats are on different sides or same side
28. (i) 136-87 m (ii) 1015.-7 m
29. 3 m 30. (1) 47-32 m (ii) 27-32 m 31. 6-83 m 32. 80 m
33. 94-64 m 34. 13-7 m 35. 2732 m

Multiple Choice Questions

1. (a) 2. (c) 3. (d) 4. (b) 5. (a) 6. (b) 7. (c)

580 UNDERSTANDING ICSE MATHEMATICS -X