41.A round balloon of radius r subtends an angle α 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 r sin β . cosec α/2.
Let P be the eye of observer. Let PA and PB are tangents to the round balloon.
PX is the horizontal line and CQ ⊥ PQ. It is given that
Let height of the centre C be h m and CA = CB = r In right triangle CBP, we have
In right triangle CPQ, we have
Hence, the height of the centre
= r sin
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Long Answer Type
42.If the angle 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 be β. Prove that the distance of the cloud from the point of observer is
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43.From an aeroplane vertically above a straight horizontal road, the angles of depression of two consecutive stones on opposite sides of the aeroplane are observed to be α and β. Show that the height in miles of aeroplane above the road is given by
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44.From the top of a lighthouse the angle of depression of two ships on the opposite sides of it are observed to be α and β. If the height of the light house be h metres and the line joining the ships passes through the foot of the lighthouse. Show that the distance
between the ship is metres.
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45.A ladder rests against a wall at an angle α to the horizontal, its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle β with the horizontal. Show that .
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46.From a window h metres 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 the opposite side of the street are ө and φ respectively. Show that the height of the opposite house is h ( 1 + tan ө. cot φ).
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47.From the top of a tower the angles of depression of two objects on the same side of the tower are found to be α and β (α > β). If the distance between the objects is p metres,
show that the height It of the tower is given by h = Also, determine the height of the tower if p = 50 metres, α = 60°, β = 30°.
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48.The angle of elevation of a cliff from a fixed point is ө. After going up a distance of K metres towards the top of the cliff at an angle of φ, it is found that the angle of elevation
is α. Show that the height of the cliff is
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49.Two stations due south of a leaning tower which leans towards the north are at distances a and b from its foot. If, α, β, are the elevations of the top of the tower from these stations,
prove that its inclination ө to the horizontal is given by cot
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50.A pole 5 m 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° and the angle of depression of the point A from the top of the tower is 45°. Find the height of the tower.
Short Answer Type
Long Answer Type
metres.
.
Also, determine the height of the tower if p = 50 metres, α = 60°, β = 30°.
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