31.The angles of depression of the top and the bottom of a 9 m high building from the top of a tower are 30° and 60° respectively. Find the height of the tower and the distance between the building and the tower.
Let AB be the building such that AB = 9 m and CD is the tower. The angles of depression of the top and the bottom of the building from the tower are 30° and 60° respectively.
In Comparing (i) and (ii), we get
⇒ 3h = h + 9 ⇒ 3h - h = 9 ⇒ 2h = 9 ⇒ h = 4.5 m Now, height of the tower = (h + 9) met. = (4.5 + 9) met. = 13.5 met. Difference between the building and tower (x)
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32.From the top of a building 60 m high the angles of depression of the top and the bottom of tower are observed to be 30° and 60°. Find the height of the tower.
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33.As observed from the top of n lighthouse, 100 m high above sea level, the angle of depression of a ship sailing directly towards it, changes from 30° to 60°. Determine the distance travelled by the ship during the period of observation.
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34.From the top of a hill, the angles of depression of two consecutive kilometre stones due east are found to be 30° and 45° respectively. Find the height of the hill.
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35.An aeroplane when flying at a height of 3000 m from the ground passes vertically above another aeroplane at an instant when the 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 aeroplane at that instant.
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36.A 7 m 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 45° and 30° respectively. Find the height of the tower.
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37.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 metres 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.
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38.From a point 100 m above a lake, the angle of elevation of a stationary helicopter is 30° and the angle of depression of reflection of the helicopter in the lake is 60°. Find the height of the helicopter.
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39.A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height y. At a point on the plane, the angles of elevation of the bottom and the top of the flagstaff are a and fi respectively. Prove that the height of the tower is
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40.If the angles of elevation of a cloud from a point h metres above a lake is α and the angle of depression of its reflection in the lake is β, prove that the height of the cloud is
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