Chapter 13 - Height and Distance

Exercise - 26

1. From the top of tower 60m high, the angles of depression of the top and bottom of a building whose base is in the same straight line with the base of the tower are observed to be 30⁰ and 60⁰ respectively. Find the height of the building.
2. An aeroplane flying horizontally at a height of .1.5 km above the ground is observed at a certain point on earth to subtend and angle of 60⁰. After 15 seconds, its angle of elevation at the same point is observed to be 30⁰. Calculate the speed of the aeroplane in km/hr.
3. A vertical tower is surmounted by a flagstaff of height h metres. At a point on the ground, the angle of elevation of the bottom and top of the flag staff are and respectively. Prove that the height of the tower is
4. If the angle of elevation of a cloud from a point h metres above a lake is and the angle of depression of its reflection in lake is , prove that the distance of the cloud from the point of observation is
5. A tower in a city is 750m high and a multi-storeyed hotel at the city centre is 50m high. The angle of elevation of the top of the tower at the top of the hotel is 30⁰. A building, h metres high, is situated on the straight road connecting the tower with the city centre at a distance the city centre at a distance of a 1.0 km from the tower. Find the value of h if the top of the hotel, the top of the building and the top of the tower are in a straight line. Also find the distance of the tower from the city centre.
6. In the adjoining figure, ABCD is a trapezium in which AB || CD. Line segments RS and LM are drawn parallel to AB such that AJ = JK = KP. If AB = O.5m and AP = BQ = 1.8m, find the length of AP, BD, RS and LM.
7. 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 find the speed of the jet plane.
8. Determine the height of a mountain if the elevation if the elevation of its top at an unknown distance from the base is 45⁰ and at a distance 10km further off from the mountain, along the same line, the angle of elevation is 30⁰ (USE tan30⁰ = 0.5774).
9. The angle of elevation of the top of a rock from the top and foot of a 100m high tower are respectively 30⁰ and 45⁰. Find the height of the rock.
10. The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60⁰. At a point Y, 40m vertically above X, the angle of elevation is 45⁰. Find the height of the tower PQ and the distance XQ.
11. An aeroplane, when 3000m high, pass vertically above another aeroplane at an instance when the angles of elevation of the two aeroplanes from the same point on the ground are 60⁰ and 45⁰ respectively. Find the vertical distance between the two aeroplans.
12. 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 60⁰ and the angle of depression of the point ‘A’ from the top of the tower is 45⁰. Find the height of the tower.
13. A tower is 50m high. Its shadow is x m shorter when the sun’s altitude is 45⁰ than when it is 30⁰. Find x.
14. From the top of a tower, the angle of depression of two objects on the same side of the tower are found to be If the distance between the object is ‘p’ metres, show that the height ‘h’ of the tower is given by .
    Also, ditermine the height of the tower if
15. A 7m long flagstaff is fixed on the top of a tower on the horizontal plane. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are 60⁰ and 45⁰ respectively. Find the height of the tower correct to one place of decimal.
16. A round balloon of radius r subtends an angle at the eye of the observer while the angle of elevation of its centre is , Prove that the height of the centre of the balloon is
17. The angle of elevation of the top of a tower as observed from a point in a horizontal line through the foot of the tower is 30⁰. When the observer moves towards the tower a distance of 100m, he finds the angle of elevation of the top to be 60⁰. Find the height of the tower and the distance of the first position from the tower.
18. The angle of elevation of the top of a tower from a point A on the ground is 30⁰. On moving a distance of 20m of towards the foot of the tower to a point B, the angle of elevation increases to 60⁰. Find the height of the tower and distance of the tower from the point A.
19. A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 30m. At a point on the plane the angle of elevation of the bottom of the flagstaff is 45⁰ and of the top of the flagstaff is 60⁰. Determine the height of the tower and the horizontal distance.
20. From the top of a tower 50m high, the angle of depression of the top and bottom of a pole are observed to be 45⁰ and 60⁰ respectively. Find the height of the pole if the pole and tower stand on the same line.
21. The angles of elevation of the top of a tower, as seen from two points A and B situated in the same line and at distances p and q respectively from the foot of the tower, are complementary. Prove that the height of the tower is .
22. From a window (60 metres high above the ground) of a house in a street the angles of elevation and depression of the top and the foot of another house on opposite side of street are 60⁰ and 45⁰ respectively. Show that the height of the opposite house is metres.
23. If the angle of elevation of a cloud from a point h metres above a lake is and the angle of depression of its reflection in the lake is . Prove that the height of the cloud is
24. From an aeroplane vertically above a straight horizontal plane, the angles of depression of two consecutive kilometer stones on the opposite sides of the aeroplane are found to be and . Show that the height of the aeroplane is .
25. A man standing on the deck of a ship, which is 10m above water level, observes the angle of elevation of the top of a hill as 60⁰ and angle of depression of the base of the hill is 30⁰. Find the distance of the hill from the ship and height of the hill.
26. As observed from the top of a light-house, 100m high above sea level, the angle of depression of a ship, sailing directly towards it, changes from 30⁰ to 60⁰. Determine the distance traveled by the ship during the period of observation .
27. The angle of elevation of a cloud from a point 200m above the 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.
28. The angle of depression of the top and bottom of a tower, as seen from the top of a 100m high cliff, are 30⁰ and 60⁰ respectively. Find the height of the tower.

Answers

1. 40m	2. 415.68 km/h
5. 172.6m, 1212.4m	6. BD = AC = 2.0785m, Rs. 1.1928m, LM = 1.8856m.
7. 720km/h	8. 13.66m
9. 236.5m	10. 94.64m, 109.3m
11. 1268m	12. 6.83m
13. 3660cm	14. 43.25m
15. 9.589m	17. 86.6m, 150m
18.17.3m, 30m	19. 40.98m, 40.98m
20. 21.13m	25. 17.3m, 40m
26. 115.46m	27. 400m,
28. 66.67m