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Some Applications of Trigonometry

Short Answer Type

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.

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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 $fraction numerator 2 straight h space sec space straight alpha over denominator tan space straight beta space minus space tan space straight alpha end fraction.$

et AB be the surface of the lake and 'P' be the position of the observer h metres above the lake. Let C be the cloud and C' be the reflection in the cloud. Then CB = C'B. It is also given that the angle of elevation of cloud from a point h m above a lake is α and angle of depression of its reflection be β. i.e., ∠CPQ = α and ∠QPC' = β. Let CQ = xm.
In right triangle PQC, we have

et AB be the surface of the lake and 'P' be the position of the obser
In right triangle PQC, we have

$sin space straight alpha space equals space CQ over CP rightwards double arrow space sin space straight alpha space equals space straight x over CP rightwards double arrow space space space space straight x space equals space CP space sin space straight alpha space space space space space space space space space space space... left parenthesis straight i right parenthesis and space space space tan space straight alpha space space equals space CP over PQ rightwards double arrow space space space tan space straight alpha space equals space straight x over PQ rightwards double arrow space space space PQ space equals space fraction numerator straight x over denominator tan space straight alpha end fraction space space space space space space space space space space space... left parenthesis ii right parenthesis$
In right triangle PQC', we have

$tan space straight beta space space equals space fraction numerator QC apostrophe over denominator PQ end fraction rightwards double arrow space space tan space straight beta space equals space fraction numerator BQ space plus space BC apostrophe over denominator PQ end fraction rightwards double arrow space space tan space straight beta space equals space fraction numerator straight h plus straight x plus straight h over denominator PQ end fraction rightwards double arrow space space space PQ space equals space fraction numerator 2 straight h plus straight x over denominator tanβ end fraction space space space space space space space space space space space space space space space... left parenthesis iii right parenthesis$
Comparing (ii) and (iii), we get

$fraction numerator straight x over denominator tan space straight alpha end fraction space equals space fraction numerator 2 straight h plus straight x over denominator tan space straight beta end fraction straight x space tan space straight beta space equals space 2 straight h space tan space straight alpha space plus straight x space tan space straight alpha rightwards double arrow space straight x space tan space straight beta space minus space straight x space tan space straight alpha space equals space 2 straight h space tan space straight alpha rightwards double arrow space straight x left parenthesis tan space straight beta space minus space tan space straight alpha right parenthesis space equals space 2 straight h space tan space straight alpha rightwards double arrow space straight x space equals space fraction numerator 2 straight h space tan space straight alpha over denominator tan space straight beta space minus space tan space straight alpha end fraction space space space space space space space space space space space space space space... left parenthesis iv right parenthesis$
Comparing (i) and (iv), we get

$CP space sin space straight alpha space equals space fraction numerator 2 straight h space tan space straight alpha over denominator tan space straight beta space minus space tan space straight alpha end fraction CP space equals space fraction numerator 2 straight h space tan space straight alpha over denominator sin space straight a left parenthesis tan space straight beta space minus space tan space straight alpha right parenthesis end fraction space space space space space equals space fraction numerator 2 straight h over denominator cos space straight alpha end fraction cross times fraction numerator sin space straight alpha over denominator sin space straight alpha end fraction cross times fraction numerator 1 over denominator left parenthesis tan space straight beta minus tan space straight alpha right parenthesis end fraction space space space space space equals space fraction numerator 2 straight h space sin space straight alpha over denominator sin space straight alpha space cos space straight alpha end fraction cross times fraction numerator 1 over denominator tan space straight beta space minus space tan space straight alpha end fraction space space space space space equals space space fraction numerator 2 straight h over denominator cos space straight alpha end fraction cross times fraction numerator 1 over denominator tan space straight beta minus tan space straight alpha end fraction space space space space space equals space space fraction numerator 2 straight h space sec space straight alpha over denominator tan space straight beta space minus space tan space straight alpha end fraction$
Hence, the distance of the cloud from the point of observer is $fraction numerator 2 straight h space sec space straight alpha over denominator tan space straight beta space minus space tan space straight alpha end fraction.$

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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

fraction numerator tan space straight alpha space. space tan space straight beta over denominator tan space straight alpha space plus space tan space straight beta end fraction.

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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

fraction numerator straight h left parenthesis tan space straight alpha space plus space tan space straight beta right parenthesis over denominator tan space straight alpha space minus space tan space straight beta end fraction

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

straight a over straight b equals fraction numerator cos space straight alpha space minus space cos space straight beta over denominator sin space straight beta space minus space sin space straight alpha end fraction

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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 =

fraction numerator straight p space tan space straight alpha. space tan space straight beta over denominator tan space straight alpha space minus space tan space straight beta end fraction

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

fraction numerator straight k left parenthesis cos space straight phi space minus space sin space straight phi. space cot space straight alpha right parenthesis over denominator cot space straight theta space minus space cot space straight alpha end fraction.

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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

straight theta space equals space fraction numerator straight b space cot space straight phi space minus space straight a space cot space straight beta over denominator straight b space minus straight a space end fraction

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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.

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