Class 10th
Application of trigonometry
Que1. 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.
Que1. 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
(see Fig. 9.5). 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 √3 = 1.73)
Que1. 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?
Que1. 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 3 = 1.732)
Que1. 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.
Que1. 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.
Que1. 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°.
Que1. 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.
Que1. 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?
Que1. 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.
Que1. 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.
Que1. 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 30° to 60° as
he walks towards the building. Find the distance he walked towards the
building.
Que1. 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.
Que1. 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.
Que1. 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.
Q16. 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.
Q17. 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.
Q18. 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.
Q19. 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.
Q20. 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.
Q1. A 1.6
m tall girl stands at a distance of 3.2m from a lamp post and casts a shadow of
4.8 m on the ground. Find the height of the lamp post
ANS. (2.6m)
Q2. A man
standing on the deck of a ship, which is 10 m above water level, observes the
angle of elevation of the top of a hill is 60˚ and the angle of depression of the base of the
hill is 30.˚ Calculate the distance of the hill from the ship and the height of
the hill.
ANS. (10√3m, 40m)
Q3. The angle of
elevation of a cloud from a point 60m above a lake is 30˚ and angle of
depression of the reflection of cloud in the Lake is 60˚. Find the height
of the cloud. ANS.
(120 m)
Q4. The angle
of elevation of a jet plane from a point
A on the ground is
60˚. After a flight of 15 sec the angle of elevation changes to 30˚. If the jet plane is flying at a constant height
of 1500√3m, then find the speed of jet plane.
ANS. (720 km /hr)
Q5. A vertical
tower stands on a horizontal
plane and is surmounted by a vertical flagstaff of height h. At
a point on the plane, the angles of elevation at the bottom and the top of the flagstaff are α and β respectively. Prove that the height of the tower
is h tan α / tanβ – tanα
Q6.
The angle of elevation of the top of a tower from two points
at distances a and b metres from the base and in the same straight line with it are complementary.
Prove that height of the tower is √ab
metres.
Q7. The angles
of elevation of the top of a rock from the top and foot of a 100 m high tower
are 30˚ and 45˚respectively. Find the
height of the rock.
ANS. (236.5 m)
Q8. A boy is standing on the ground and is flying a kite with
100m of string at an elevation of 30˚ Another
boy is standing on the roof of a 10m
high building and is
flying his kite at an elevation of 45˚. Both the boys are on opposite sides of the kite’s .Find
the length of the string that the Second boy must
have so that two kites meet.
ANS. (40√2 m)
Q9. 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.
ANS. (20√3m)
Q10. The angle of elevation ø of a vertical tower from a point on ground is such
that its tangent is 5/12. On walking
192m towards the tower in the same straight line, the tangent
of the angle of elevation Is found to be ¾. Find the height of the tower
ANS. (180 m)
Q11. A bird is
sitting on the top of a tree, which is 80m high. The angle of elevation of the
bird, from a point on the ground is 45˚. The bird flies away from the point of
observation horizontally and remains at a Constant height. After 2 sec, the
angle of Elevation of the bird from
the point of observation becomes 30˚. Find
the speed of flying of the bird
ANS. (29.28m/sec)
Q12. An aero
plane at an altitude of 200m observes the angles of depression of opposite
points on the two banks of a river to be 45˚ and 60˚. Find the width of the
river ANS. (315.4m)
Q13. Two men on
either side of a cliff, 60m high, observe the angles of elevation of the top of
the cliff to be 45˚ and 60˚ respectively Find the distance between two men ANS.
(94.6m)
Q1.
A vertical stick 10 cm long casts a shadow 8 cm long. At the same time, a tower
casts a shadow 30 m long. Determine the height of the tower.
Q2. An
observer, 1.5 m tall, is 28.5 m away from a tower 30 m high. Find the angle of elevation
of the top of the tower from his eye.
Q3. A person standing
on the bank of a river observes that the angle subtended by a tree on the opposite
bank is 60o. When he retreats 20m from the bank, he finds the angle to be 300.
Find the height of the tree and the breadth of the river.
Q4. A
boy is standing on ground and flying a kite with 150m of string at an elevation
of 300. Another boy is standing on the roof of a 25m high building and flying a
kite at an elevation of 45o. Find the length of string required by the second
boy so that the two kites just meet, if both the boys are on opposite side of
the kites.
Q5. An
aeroplane flying horizontally 1000m above the ground, is observed at an angle
of elevation 600 from a point on the ground. After a flight of 10 seconds, the angle
of elevation at the point of observation changes to 300. Find the speed of the
plane in m/s.
Q6. An
aeroplane when flying at a height of 4000 m from the ground passes vertically above
another aeroplane at an instant when the angles of the elevation of the two
planes from the same point on the ground are 600 and 450 respectively. Find the
vertical distance between the aeroplanes at that instant.
Q7. An
aeroplane at an altitude of 200 m observes the angles of depression of opposite
points on the two banks of a river to be 450 and 600. Find the width of the
river.
Q8. The
shadow of a flag staff is three times as long as the shadow of the flag staff when
the sun rays meet the ground at an angle of 600. Find the angle between the sun
rays and the ground at the time of longer shadow.
Q9. A
vertically straight tree, 15m high is broken by the wind in such a way that it
top just touches the ground and makes an angle of 600 with the ground, at what
height from the ground did the tree break?
Q10. A
man in a boat rowing away from lighthouse 100 m high takes 2 minutes to changes
the angle of elevation of the top of lighthouse from 600 to 450. Find the speed
of the boat.
Q11. As observed from the top of a light house, 100m
above sea level, the angle of depression of ship, sailing directly towards it, changes
from 300 to 450. Determine the distance travelled by the ship during the period
of observation.
Q12.
A man standing on the deck of ship, which is 10m above the water level, observes
the angle of elevation of the top of a hill as 600 and the angle of depression
of the base of the hill as 300. Calculate the distance of the hill from the
ship and the height of the hill.
Q13.
The angles of elevation of the top of a tower from two points at a distance of
‘a’ m and ‘b’ m from the base of the tower and in the same straight line with
it are complementary, then prove that the height of the tower is √ab
Q14.
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.
Q15.
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?
Q16.
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?
Q17.
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.
Q18.
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.
Q19.
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.
Q20.
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.
Q21.
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 30° to 60°
as he walks towards the building. Find the distance he walked towards the
building.
Q22.
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.
Q23.
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.
Q24.
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.
Q25.
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 60°. After some time, the angle of elevation
reduces to 30°. Find the distance travelled by the balloon during the interval.
Q26.
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.
Q27.
A man on cliff observes a boat an angle of depression of 300 which is approaching
the shore to the point immediately beneath the observer with a uniform speed.
Six minutes later, the angle of depression of the boat is found to be 600. Find
the time taken by the boat to reach the shore.
Q28.
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.
Q29.
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.
Q30.
A tree is broken by the storm. The top of the tree touches the ground making an
angle 30° and at a distance of 30 m from the root. Find the height of the tree.
Q31.
A tree 12m high, is broken by the storm. The top of the tree touches the ground
making an angle 60°. At what height from the bottom the tree is broken by the
storm.
Q32.
At a point on level ground, the angle of elevation of a vertical tower is found
to be such that its tangent is 12 . In walking 192 m towards the tower, the tangent
of the angle of elevation is 4 . Find the height of the tower.
Q33.
The pilot of an aircraft flying horizontally at a speed of 1200km/hr, observes
that the angle of depression of a point on the ground changes from 300 to 450
in 15 seconds. Find the height at which the aircraft is flying.
Q34.
If the angle of elevation of the cloud from a point h m above a lake is A and
the angle of depression of its reflection in the lake is B, prove that the
height of the cloud is h(tanB ttan A)
Q35.
The angle of elevation of cloud from a point 120m above a lake is 300 and the angle
of depression of the reflection of the cloud in the lake is 600. Find the height
of the cloud.
Q36.
The angle of elevation of cloud from a point 60m above a lake is 300 and the angle
of depression of the reflection of the cloud in the lake is 600. Find the height
of the cloud.
Q37.
The angle of elevation of a jet plane from a point A on the ground is 600. After
a flight of 15 seconds, the angle of elevation changes to 300. If the jet plane
is flying at a constant height of 1500 3 m, find the speed of the jet plane.
Q38.
The angle of elevation of a jet plane from a point A on the ground is 600. After
a flight of 30 seconds, the angle of elevation changes to 300. If the jet plane
is flying at a constant height of 3600 3 m, find the speed of the jet plane.
Q39.
There are two temples, one on each bank of river, just opposite to each other.
One temple is 50m high. From the top of this temple, the angles of depression
of the top and foot of the other temple are 300 and 600 respectively. Find the
width of the river and the height of other temple.
Q40.
A ladder rests against a wall at an angle α to the horizontal, its foot is
pulled away from the wall through a distant a, so that it slides a distance b
down the wall making an angle β with the horizontal. Show that b sinβsinα .
Q41.
From a window, h meter above the ground of a house in a street , the angle of elevation
and depression of the top and the foot of another house on the opposite side of
the street are and respectively. Show that the height of the opposite house
is h (1 + tancot).
Q42.
From a window, 15 meters high above the ground of a house in a street , the angle
of elevation and depression of the top and the foot of another house on the
opposite side of the street are 300 and 450 respectively. Find the height of the
opposite house.
Q43. Two
stations due south of a leaning tower which leans towards the north are at distances
a and b from its foot. If α and β are the elevations of the top of the tower
from these stations, prove that its inclination θ to the horizontal is given
bycotθ bcotαacotβ .
Q44.
The angle of elevation of a cliff from a fixed point is θ. After going up a
distance of ‘k’meters towards the top of the cliff at an angle of , it is
found that the angle of elevation is α. Show that the height of the cliff is
k(cossin.cotα) .
Q45.
A round balloon of radius r subtends an angle α at the eye of the observer while
the angle of elevation of its centre is β. Prove that the height of the centre
of the balloon is rsinβ.cosec 2
Q46.
The angle of elevation of the top of a tower from a point on the same level as
the foot of the tower is α. On advancing ‘p’ meters towards the foot of the tower
the angle of elevation becomes β. Show that the height ‘h’ of the tower is
given by h = tanβtanα m. Also determine the height of the tower if p = 150o
m, α = 30o and β = 60o.
Q47.
From the top of a light- house the angle of depression of two ships on the
opposite sides of it are observed to be α and β. If the height of the light-house
be ‘h’ meter and the line joining the ships passes through the foot of the light
house, show that the distance between the ships is tanαtanβ tanα.tanβ
Q48.
An electrician has to repair on electric fault on a pole of height 4m. she needs
to reach a point 1.3m below the top of the pole to undertake the repair work.
What should be the height of the ladder that she should use at angle of 60o to
the horizontal, would enable her reach the required position? Also, how far the
foot of the pole should she place the foot of the ladder.( take 3 = 1.732)
Q49.
The angle of elevation of a jet fighter from a point A on the ground is 60o.
After a flight of 15 sec, the angle of elevation changes to 30o. If the jet is
flying at a speed of 720 km/hr, find the constant height at which the jet is flying.
Q50.
A man on a top of a tower observes a truck at angle of depression α where tanα
=5 and sees that it is moving towards the base of the tower. Ten minutes later,
the angle of depression of truck found to be β where tanβ = 5 if the truck is
moving at uniform speed determine how much more time it will take to reach the
base of the tower.
Q51.
At the foot of a mountain the elevation of its summit is 450; after ascending
1000m towards the mountain up a slope of 300 inclination, the elevation is
found to be 600. Find the height of the mountain.
Q52.
If the angle of elevation of cloud from a point h metres above a lake is α and
the angle of depression of its reflection in the lake be β, prove that the distance
of the cloud from the point of observation is tanβtanα
Q53.
A vertical tower stands on a horizontal plane and is surmounted by a vertical
flag staff of height ‘h’. At a point on the plane, the angles of elevation of
the bottom and top of the flag staff are and respectively. Prove that the
height of the tower is tantan.
Q54.
A man on the top of a vertical tower observes a car moving at a uniform speed coming
directly towards it. If it takes 12 minutes for the angle of depression to change
from 300 to 450, how soon after this, will the car reach the tower? Give your answer
to the nearest second.
Q55.
Two pillars of equal height and on either side of a road, which is 100m wide.
The angles of depression of the top of the pillars are 600 and 300 at a point
on the road between the pillars. Find the position of the point between the
pillars and the height of the tower.
Q56.
The angle of elevation of the top of a tower from a point A due north of the
tower is and from B due west of the tower is . If AB = d, show that the
height of the tower is sini
sin2 .
Q57.
The angle of elevation of the top of a tower from a point A due south of the
tower is and from B due east of the tower is . If AB = d, show that the height
of the tower is cot2 cot2 .
Q58.
From an aeroplane vertically above a straight horizontal road, the angles of depression
of two consecutive milestones on opposite sides of the aeroplane are observed
to be and . Show that the height in miles of aeroplane above the road is
given by tanαtanβ .
Q59.
A tree standing on horizontal plane is leaning towards east. At two points situated
at distances a and b exactly due west on it, angles of elevation of the top are
respectively and . Prove that the height of the top from the ground is (ba)tanαtanβ
.
Q60.
The length of the shadow of a tower standing on level plane is found to be 2x
metres longer when the sun’s altitude is 300 than when it was 450. Prove that
the height of tower is x 3 1 m .
