CBSE Guess > Papers > Important Questions > Class X > Maths > Maths (Heights And Distances) By Mr. Tanmaya Tripathi Heights And Distances 1. Two ships are sailing in the sea on either side of a lighthouse. The angles of depression of the two ships are observed as 600 and 450 respectively. If the distance between the two ships is 2. From the top of a lighthouse , the angles of depression of two ships on the opposite sides of it are observed to be ά and β. If the height of the light house be h meters and the line joining the ships passes , through the foot of the lighthouse, show that the distance between the ships is h(tan α +tanβ) tanα.tanβ Or A round balloon of radius r subtends an angle ά 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 rsin β cosec α/2 3. An airplane when flying at a height of 4000 m from the ground passes vertically above another airplane at an instant when the angles of elevation of the two airplanes from the same point on the ground are 60 and 45 respectively. Find the vertical distance between the two airplanes at that instant. Or The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60. At a point Y, 40m vertically above X, the angle of elevation is 45. Find the height of the tower PQ and the distance XQ. 4. The angle of elevation of a cliff from a fixed point is Or The angle of elevation of a jet plane from a point A on the ground is 60 . After a flight of 15 seconds , the angle of elevation changes to 30 , If the jet plane is flying at a constant height height of 1500√3 m , find the speed of the jet plane 5. 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 30 0 to 45 0 , how soon after this, will the car reach the tower? Give your answer to the nearest second. Or The angle of elevation of the top of a tower from a point on the same level as the foot of the tower is ά . On advancing p meters towards the foot of the tower, the angle of elevation become β . Show that the height of the tower is 6. The hypotenuse of a right triangle is 1m less than twice the shortest side. If the third side is 1m more than the shortest side, find the sides and area of the triangle. Or A swimming pool is filled with three pipes with uniform flow. The first two pipes operating simultaneously fill the pool in the same time during which the pool is filled by the third pipe alone. The second pipe fills the pool five hours faster than the first pipe and four hours slower than the third pipe. Find the time required by each pipe to fill the pool separately. 7. A pole 5m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point ‘A’ on the ground is 60o and the angle of depression of the point ‘A’ from the top of the tower is 45o. Find the height of the tower. 8. 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 moves 30 m away from the bank, he finds the angle of elevation to be 30 o. Find the height of the tree and the width of the river. Or The angle of elevation of a jet plane from a point A on the ground is 60 o .After flight of 7 seconds, the angle of elevation changes to 30 o . If the jet is flying at a constant height of 1750√3 m. find the speed of the jet plane. 9. 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 be the car reach the observation tower. 16.4 min appro Or An aircraft is flying along a horizontal course AB directly towards an observer on the ground at P, maintaining an altitude of 5000 m. when the aircraft is at A, the angle of depression is 30 o and when at B ,it is 60 o respectively. Calculate the distance AB. 5773.33m 10. 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 O. If the height of the second tower is 60 m, find the height of the first tower. 11. Two stations due south of a leaning tower which leans towards north are at distances a & b from its foot. If angles of elevation of top of tower are α&β from these stations. Prove that inclination θ to the horizontal is given by cot Ø= (b cotα – a cotβ) / (b-a). 12. From the top of a lighthouse, the angles of depression of two ships on the opposite sides of it are observed to be ά and β. If the height of the light house be h meters and the line joining the ships passes, through the foot of the lighthouse, show that the distance between the ships is 13. An airplane flying horizontally at a height of 1.5km above the ground is observed at a certain point on earth to subtend an angle of 60 o . After 15 seconds, its angle of elevation at the same point is observed to be 30°. Calculate the speed of the aero plane in km/h. 14. The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle of depression of the reflection of cloud in the lake is 60°. Find the height of the cloud. 15. The angle of elevation of a tower as seen from a point ‘A’ due North of it is ‘α’ and that as seen from a point B due East of A is ‘β’. Prove that the height of the tower is AB sinα sinβ/√sin2α – sin2β 16. A round balloon of radius r subtends an angle α at the eye of the observer while the angle of elevations of its centre is β . Prove that the height of the centre of the balloon is Or The angle of elevation of a cloud from a point 60m above a lake is 300 and the angle of depression of the reflection of cloud in the lake is 600. Find the height of the cloud. 17. 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 time taken by the boat to reach the shore Or The angles of elevation of the top of a tower from two points P and Q at distances of a and b respectively, from the base and in the same straight line with it are complementary. Prove that the height of the tower is √ab 18. A pole 5m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point ‘A’ on the ground is 60 o and the angle of depression of the point ‘A’ from the top of the tower is 45 o. Find the height of the tower. 19. A man on the cliff observes a boat at an angle of depression of 30 approaching the shore immediately beneath the observer with uniform speed . Six minutes later the angle of depression of that boat is found to be 60 .Find the time taken by the boat to reach the shore Or A bird sitting on the top of a tree which is 80 m 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 constant height. After 2 seconds the angle of elevation of the bird from the point of observation becomes 30. Find the speed of the flying bird 20. The angle of elevation of a cloud from a point 200m above the lake is 300 and the angle of depression of its reflection in the lake is 60 0. Find the height of the cloud. Or A man on a cliff observes a boat at angle of depression of 300 which is approaching the shore to the point immediately beneath the observer with 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. 21. A pole is projected outwards from a window 10 m above the ground of a building makes an angle of 300 with the wall. The angles of elevations of the bottom and top of the pole from a point on the ground are 300and 600 respectively. Find the length of the pole. |
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Find the height of the lighthouse. 
. After going up a distance of k meters towards the top of the cliff at an angle of 
, it is found that angle of elevation is ά .Show that the height of the cliff is 
meters.
. Also, determine the height of the tower if p = 150 meters, ά = 30 o and β = 60 o .
