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Some Applications of Trigonometry

Long Answer Type

181.

The horizontal distance between two poles is 15 m. The angle of depressionof the top of first pole as seen from the top of second pole is 30o. If theheight of the second pole is 24 m, find the height of the first pole. $|Us e \sqrt{3} = 1.732|$

182.
Two poles of equal heights are standing opposite each other on either sideof the roads, which is 80 m wide. From a point between them on theroad, the angles of elevation of the top of the poles are 60o and 30orespectively. Find the height of the poles and the distances of the pointfrom the poles.

$Let AC and BD be the two poles of the same height h m . Given : AB = 80 m Let AP = x m, therefore, PB = (80 - x) m In ∆ APC, \tan 30 ° = \frac{AC}{AP} \frac{1}{\sqrt{3}} = \frac{h}{x} . . . . . . . . . (1) In ∆ BPD, \tan 60 ° = \frac{BD}{AB} \sqrt{3} = \frac{h}{80 - x} . . . . . . . . . . . . . (2) Dividing (1) by (2), \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}} = \frac{\frac{h}{x}}{\frac{h}{80 - x}} \Rightarrow \frac{1}{3} = \frac{80 - x}{x} \Rightarrow x = 240 - 3 x \Rightarrow 4 x = 240 \Rightarrow x = 60 From (1), \frac{1}{\sqrt{3}} = \frac{h}{x} \Rightarrow h = \frac{60}{\sqrt{3}} = 20 \sqrt{3} m Thus, the height of both the poles is 20 \sqrt{3} m and the distance of the point from the poles are 60 m and 20 m .$

183.

The angle of elevation of an aeroplane from a point on the ground is 60°. After a flight of 30 seconds the angle of elevation becomes $30 °$ . If the aeroplane is flying at a constant height of 3000 $\sqrt{3}$ m, find the speed of the aeroplane.

184.

The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is 45°. If the angle of elevation of the top of a flagstaff fixed at the top of the tower, at A is 60°, then find the height of the flagstaff. [Use $\sqrt{3} = 1.73$ ]

Multiple Choice Questions

185.

The angle of depression of a car parked on the road from the top of a 150 m high tower is 30°. The distance of the car from the tower (in metres) is

$50 \sqrt{3}$
$150 \sqrt{3}$
$150 \sqrt{2}$
75

Short Answer Type

186.

If a tower 30 m high, casts a shadow 10 3m long on the ground, then what is the angle of elevation of the sun?

Long Answer Type

187.

On a straight line passing through the foot of a tower, two points C and D are at distances of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then find the height of the tower.

188.

From the top of a tower, 100 m high, a man observe two cars on the opposite sides of the tower and in same straight line with its base, with its base, with angles of depression 30° and 45°. Find the distance between the cars.
[Take $\sqrt{3}$ = 1.732]

Multiple Choice Questions

189.

The length of shadow of a tower on the plane ground is 3 times the height of the tower. The angle of elevation of sun is

45⁰
30⁰
60⁰
90⁰

Long Answer Type

190.

A kite is flying at a height of 45 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60o. Find the length of the string assuming
that there is slack in the string.