class 10 Application of trigonometry solved Board Questions

Class 10 Application of Trigonometry [Height and Distance] Solved Problems (14Q)

Question 01: The angle of elevation of an aeroplane from a point on the ground is 45 o. After a flight of 15 seconds, the elevation changes to 30 o. If the aeroplane is flying at a constant height of 3000 meters, find the speed of the plane.

Questions 02: A around balloon of radius r subtends an angle a at the eye of the observer while the angle of elevation of its centre is b , prove that the height of the centre of the balloon is r sin b cosec a/2

Assignment SA-2 Class X Topic : Height And Distance

1.The angle of elevation of a ladder leaning against a wall is 60^o and the foot of the ladder is 9.5 meter away from  the wall. Find the length of the ladder. [ 19m ]

2. If the length of the shadow cast by a pole be  times the length of the pole, find the angle of elevation of the sun. [ 30^o ]

3.   A tree is broken by the wind. The top stuck the ground at an angle of 30^o and at a distance of 30 m from the root.   Find the total height of the tree.

4.     A circus artist is climbing from the ground along a rope stretched from the top of vertical pole and tied at the ground level 30^o. Calculate the distance covered by the artist in climbing to the top of the pole. [ 24 m ]

5. A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60^o. When he was 40 m away from the bank he finds that the angle of elevation to be 30^o.  Find: - 

(i) The height of the tee, 

(ii) The width of the river, correct up to two decimal places.[(i)34.64m (ii) 20m]

6.  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 elevations of two planes to a same point on the ground are 60^o and 45^o respectively.  Find the vertical distance between the aeroplanes at that at that instant. [ 1693.34 m ]

7.  The angle of elevation of the top of the hill at the foot of a tower is 60^o and the angle of elevation of the top of tower from the foot of hill is 30^o. If the tower is 50 m high, what is the height of the hill. [ 150 m ]

8.    There is a small island in the middle of a 100 m wide river and a tall tree stands on the island. Let P and Q be       points directly opposite each other on the two banks, and in line with the tree. If the angles of elevation of the top     the tree from P and Q respectively are 30^o and 45^o, find the height of the tree. 

9.        Two pillars of equal heights are on either sides of a roadway, which is 150 m wide. The angles of elevation of       the top of pillars are 60^o  and 30^o at a point on the roadway between the pillars. Find the position of the point    between the pillars and the height of each pillar.       (64.95m)

10.        At the foot of mountain, the elevation of its peak is 45^o. After ascending 1 km towards the mountain up an inclination of 30^o, the elevation changes to 60^o. Find the height of mountain. (1.366 km)

11.        From the top of the building 15m high, the angle of elevation of the top of a tower is found to be 30^o. From the       bottom of the same building, the angle of elevation of the top of tower is found to be 30^o. Find the height of the     tower and the distance between the tower and the building. (22.5m, 12.975m)

12.        A fire in a building B is reported on the telephone to two fire stations P and Q, 10 km apart from each other on a    straight road. P observes that the fire is at angle of 60^o to the road and Q observe that it is an angle of 45^o to the       road. Which station should send its team and how much this team has to travel? (P, 7.32km)

13.        The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sunrays meet the ground  at an angle of 60^o. Find the angle between the sunrays and the ground at the time of long shadow. ( 30^o)

14.        From a point in the cricket ground, the angle of elevation of a vertical tower is found to be θ at a distance of  200m from the tower. On walking 125 m towards the tower the angle of elevation becomes 2θ. Find the height of   tower. (100m)

15.        A boy standing on the ground and flying a kite with 75 m of string at an elevation of 45^o. Another boy is standing on the roof of 25 m high building and is flying his kite at an elevation of 30^o. Both the boys are on the opposite side of the two kites. Find the length of the string that the second boy must have, so that the kites meet.(56.05 m)

16.        As observed from the top of light house, 100m high above the sea level, the angle of depression of a ship, sailing directly towards it, changes from 30^o to 45^o. Determine the distance traveled by the ship during the period of observation.  ( 73.2m)

17.        An aeroplane at an altitude of 200 m observes the angle of depression of opposite points on two banks of a river  to be 45^o and 60^o. Find the width of the river.  ( 315.4m)

18.        From the top of a cliff 150m high, the angles of depression of two boats are 60^o and 30^o. Find the distance  between the boats, if the  boats are (i) on the side of cliff. (ii) on the opposite sides of the cliff.     [ (i) 173.2m (ii) 346.4m ]

19.        A man standing on the deck of a ship, which is 10m above the water level, observe the angle of elevation of the  top of a hill as 60^o and the angle of depression of the base of the hill as 30^o.Calculate the distance of the hill from the ship and the height of the hill.   [17.3m, 40m].

20.        The angle of elevation and depression of the top and the bottom of a light house from the top of the building, 60m high, are 30^o and 60^o respectively. Find (i) The difference between the heights of the light house and the  building (ii) Distance between the light house and the building. [ (i) 20m, (ii) 34.64m]

21.A pole 5m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from Point ‘A’ on the ground is 60^o and the angle of depression of the point ‘A’ from the top of tower is 45^o. Find the height of tower.  [ 6.83m]

22.        Man on a cliff observes a boat at an angle of depression of 30^o 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 60^o. Find the time taken by the boat to reach the shore. [ 9 minutes]

23.  A man on the top of a vertical observation 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 30^o to 45^o, how soon after this will       the car reach the observation tower. Give your answer correct to nearest seconds.  [16 min. 24 sec.]

                                                     JSUNIL TUTORIAL CBSE MATHS & SCIENCE

ASSIGNMENT CLASS X APPLICATIONS OF TRIGONOMETRY

1. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with the ground. The distance from the foot of the tree to the point where the top touches the ground is 10 m. Find the height of the tree before it was broken.

2. From the top of a building 60 m high the angles of depressions of the top and the bottom of a tower are observed to be 30° and 60°. Find the height of the tower.

3. From a point on the ground the angles of elevation of the bottom and top of a water tank kept at the top of 20 m high tower are 45° and 60°. Find the height of the water tank.

4. A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60°. When he moves 40 meters away from the bank, he finds the angle of elevation to be 30°. Find the height of the tree and the width of the river.

5. The angle of elevation of the top of a tower from a point A on the ground is 30°. On moving a distance of 20 metres towards the foot of the tower to a point B, the angle of elevation increases to 60°.Find the height of the tower and the distance of the tower from the point A.

6. A flagstaff stands on the top of a 5 m high tower. From a point on the ground, the angle of elevation of the top of the flag-staff is 60° and from the same point, the angle of elevation of the top of the tower is 45°. Find the height of the flag-staff.

7. The shadow of a tower, when the angle of elevation of the sun is 45°, is found to be 10 m longer than when it was 60°. Find the height of the tower.

8. On a horizontal plane there is a vertical tower with a flag pole on the top of the tower. At a point 9metres away from the foot of the tower the angle of elevation of the top and bottom of the flag pole are60° and 30° respectively. Find the height of the tower and the flag pole mounted on it.

9. A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from him at an elevation of 30°. A girl standing on the roof of 20 metre high building, finds the angle of elevation of the same bird to be 45°. Both the boy and the girl are on opposite sides of the bird. Find the distance of bird from the girl.

10. From the top of a building 15 m high the angle of elevation of the top of a tower is found to be 30°.From the bottom of the same building, the angle of elevation of the top of the tower is found to be 60°.Find the height of the tower and the distance between the tower and the building.

11.At a point on the level ground the angle of elevation of a vertical tower is found to be such that its tangent is 5/12. On walking 192 m towards the tower, the tangent of the angle is found to be3/4 . Find the height of the tower.

12. Two men on either side of a cliff 80 m high observe the angles of elevation of top of the cliff to be 30° and 60° respectively. Find the distance between the two men.

13. An aeroplane, when 1500 m high passes vertically above another aeroplane at an instance when the angles of the two aeroplanes from the same point on the ground are 60° and 45° respectively. Find the vertical distance between the two aeroplanes.

14. The angles of elevation of the top of a tower from two points on the ground at distances 9m and 4m from the base of the tower are in same straight line with it are complementary. Find height of the tower.

15. 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.

APPLICATION OF TRIGONOMETRY(Height and Distance) 10th maths test paper

Math Adda 

Q1 A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60°with the wall, find the height of the wall.                                                                                  (7.5 √3 )

Q 2  A pole 12 m high casts a shadow 4 √3 m long on the ground. Find the angle of elevation        (60°)

Q 3 The angle of elevation of the top of a tower from a point on the ground is 30° if on walking 30m towards the tower, the angle of elevation becomes 60°.Find the height of the tower.(15√3 )

Q 4 An observer 1.5m tall is 20.5m away from a tower 22m high. Determine the angle of elevation of the top of the tower from the eye of the observer.                                       (45°)

Q 5 An aero plane when flying at a height of 5000m from the ground passes vertically above another aero plane at an instant when the angles of the elevation of the two planes from the same point on the ground are 60° and 45° respectively. Find the vertical distance between the aero planes at the instant.                                                                                           (2116.5m)