Class 10 Height and Distance Question : powerful question

Class 10 Height and Distance Question

Q.1. A vertical pole stands on the level ground . From a point on the ground, 25 m away from the foot of the pole, the angle of elevation of its top is found to be $60\degree$ . Find the height of the pole, [ $\sqrt{3} = 1.732$ ]

ANS :- 43.3

Q.2. A kite is flying , attached to a thread which is 165 cm long . The thread makes an angle of $30\degree$ with the ground . Find the height of the kite from the ground , assuming that there is no slack in the thread .

ANS :- 82.5

Q.3. The length of a string between a kite and a point on the ground is 85 m . If the string makes an angle $\theta$ with the ground level such that tan $\theta = \dfrac{15}{8}$ then find the height of the kite from the ground . Assume that there is no slack in the string .

ANS :- 75

Q.4. 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

ANS :- 7.5 meters

Q.5. If a tower 30 m high, casts a shadow 10√3 m long on the ground, then what is the angle of elevation of the sun?

ANS :- $\theta = 60\degree$

Q.6. If a 1.5-m-tall girl stands at a distance of 3 m from a lamp-post and casts a shadow of length 4.5 m on the ground then find the height of the lamp-post.

ANS :- 1.5 METERS

Q.7. The shadow of a tower, when the angle of elevation of the sun is 45°, is found to be 10 metres longer than when the angle of elevation is 60°. Find the height of the tower. [Given √3=1.732.]

ANS :- 23.66

Q.8. A tower is 50 m high. Its shadow is x metres shorter when the sun’s altitude is 45° than when it is 30°. Find the value of z [Given √3=1.732]

ANS :- X = 36.6

Q.9. The shadow of a tower standing on a level ground is found to be 30m longer when the sun’s altitude is 30°, than when it was 60°. Find the height of the tower. [Take √3=1.732]

ANS :- 25.98

Q.10. From a point O on the ground, the angle of elevation of the top of a tower is $30\degree$ and that of the top of the flagstaff on the top of the tower is $60\degree$ . If the length of the flagstaff in 5 meters, find the height of the tower.

ANS :- 2.5

Q.11. The angle of elevation of an aeroplane from a point on the ground is 60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the aeroplane is flying at a constant height of 1500/3 m, find the speed of the plane in km/hr.

ANS :- 720 KM/hr

Q. 12. The angles of elevation and depression of the top and bottom of a tower from The top of a building-60 m high are 30° and 60 respectively. Find the difference between the heights of the building and the tower and also the distance between them.

ANS :- $20\sqrt{3}$ m

Q. 13. A man on a cliff observes a boat at an angle of depression of 30° 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°. Find the total time taken by the boat to reach the shore.

ANS :- 9 minutes

Q. 14. Observed from the top of a 75-m -high lighthouse(from 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 ships. (Use√3= 1.73)

ANS :- X = 54.75

Q. 15. Two ships are approaching a lighthouse from opposite directions. The angles of depression of the two ships from the top of a lighthouse are 30° and 45°. If the distance between the two ships is 100 metres , find the height of the lighthouse. (Use √3=1.732.)

ANS :- 36.6

Q. 16. From the top of a lighthouse, the angles of depression of two ships on the opposite sides of it are observed to be a and B. If the height of the lighthouse be h metres and the line joining the ships passes through the foot of the lighthouse, show that the distance between the ships is $\dfrac{h(tan \propto + tan ẞ)}{tan \propto tan ẞ}$ meters.

ANS :- $\dfrac{h(tan \propto + tan ẞ)}{tan \propto tan ẞ}$ meter

Q. 17. The angle of elevation of a cloud from a point h metres above a lake is a and the angle of depression of its reflection in the lake is ẞ. Prove $\dfrac{h(tan ẞ + tan\propto )}{(tan ẞ – tan\propto )}$ that the height of the cloud is metres.

ANS :- $\dfrac{h(tan\propto + tanẞ )}{(tan ẞ – tan\propto )}$ metres.

Q. 18. Two pillars of equal heights stand on either side of a road which is 100 m wide. At a point on the road between the pillars, the angles of elevation of the tops of the pillars are 60° and 30°. Find the height of each pillar and position of the point on the road. [Take √3=1.732]

ANS :- H = 43.3

Q. 19. The angle of elevation of a cloud from a point 60 m above the surface of the water of a lake is 30° and the angle of depression of its shadow in water of lake is 60°. Find the height of the cloud from the surface of water.

ANS :- 120 Metres.

Q.20 . A round balloon of radius r subtends an angle a at the eye of the observer while the angle of elevation of its centre is ẞ. Prove that the height of the centre of the balloon is $r sinẞ cosec \dfrac{\propto}{2}$ .

ANS :- $r sinẞ cosec \dfrac{\propto}{2}.$

Q.21. A boy whose eye level is 1.3 m from the ground, spots a balloon moving with the wind in a horizontal line at some height from the ground. The angle of elevation of the balloon from the eyes of the boy at an instant is 60°. After 2 seconds, the angle of elevation reduces to 30°. If the speed of the wind is 29/3 m/s then find the height of the balloon from the ground.

ANS :- 88.3 meters

Q.22. An aeroplane when flying at a height of 3000 metres from the ground LCD and A passes vertically above another aeroplane at an instant when the EBD=60 angles of elevation of the two planes from the same point on the ground are 60° and 45° respectively. Find the vertical distance between the aeroplanes at that instant. [Take √3= 1.73.]

ANS :- 1270 meters

Q.23. A man standing on the deck of a ship, which is 10 m above the water level, observes the angle of elevation of the top of a hill as 60°, and the angle of depression of the base of the hill as 30. Find the distance of the hill from the ship and the height of the hill.

ANS :- 40 meters

Q.24. From a window (h metres high above the ground) of a house in a street, the angles of elevation and depression of the top and the foot of C another house on the opposite side of the street are $\theta and \phi$ and respectively. Show that the height of the opposite house is $h(1+tan \theta cot \phi )$ metres.

ANS :- $h(1+tan \theta cot \phi )$ metres.

Q. 25. From the top of a building 60 m high, the angles of depression of the top and bottom of a tower are observed to be 30° and 60°. Find the height of the tower

ANS :- 40 M

Q. 26. The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 30 seconds, the angle of elevation changes to 30°. If the jet plane is flying at a constant height of 3600√3 metres, find the speed of the jet plane.

ANS :- 864 KM/HOUR

Q.27. The angle of elevation of a jet fighter from a point A on the ground s 60°. After a flight of 15 seconds, the angle of elevation changes to 30 If the jet is flying at a speed of 720 km/hour, find the constant ON height at which the jet is flying. [Use √3 = 1.732.]

ANS :- 2598 M.

Q. 28. A 1.2-m-tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 metres 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.

ANS :- $58\sqrt{3}$ m

Q.29. A tree breaks due to storm and the broken part bends so 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 th ground is 9 m. Find the height of the tree.

ANS :- $9\sqrt{3} m$

Q. 30 . 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 30 metres away from the bank, he finds the angle at elevation to be 30°. Find the height of the tree and the width a the river. [Take √3=1.732.].

ANS :- The height of tree is 25.98 m and the width of the river is 15 metres.

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