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Some Applications of Trigonometry

Short Answer Type

161. A pole of height 5 m is fixed on the top of a tower The angle of elevation of the top of the pole as 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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162. 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.

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163. A man on the deck of a ship 14 m above water level, observes that the angle of elevation of the top of a cliff is 60° and the angle of depression of the base of the cliff is 30°. Calculate the distance of the cliff from the ship and the height of the cliff.

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164. In the following Fig. what are the angles of depression from the observing positions O₁ and O₂ of the object at A?

(a) 30°, 45° (b) 30°, 60° (c) 45°, 60° (d) 45°, 45°

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165.

Find the angle of elevation of the sun's altitude when the height of shadow of a vertical pole is equal to its height.
(a) 30° (b) 60° (c) 45° (d) 90°

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166. The string of a kite is 100 m long and its makes an angle of 60° with the horizontal. Find the height of the kite. Assume that there is no slackness in the string.

(a) 50 m (b) 60 m (c)

50 over 3 space straight m

(d)

50 square root of 3 space straight m space

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167.

In Fig. 9.74, what are the angles of depressions of the top and bottom of a pole from the top of a tower h m high.
(a) 45°, 70° (b) 45°, 60° (c) 45°, 30° (d) 45°, 45°

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168. A balloon is connected to a meterological ground station by a cable length 215 m inclined at 60° to the horizontal determine the height of the balloon leave the ground. Assume the there is no slack in the cable.

left parenthesis straight a right parenthesis space space 215 over 2 straight m space space space space space space space space space space left parenthesis straight b right parenthesis space 215 square root of 3 straight m space space space space space space space space space space left parenthesis straight c right parenthesis space fraction numerator 215 square root of 3 over denominator 2 end fraction straight m space space space space space space space left parenthesis straight d right parenthesis space 215 space straight m

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169.

The angle of elevation of the top of a tower from a point on the grounds which is 30 m away from the foot of the tower is 30°. The height of the tower is

(a) 10 m (b) $fraction numerator 10 over denominator square root of 3 end fraction straight m$ (c) $10 square root of 3 space straight m$ (d) 30 m

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170.
A ladder leaning against a wall makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.

Let AB be the ladder and CA be the wall.

The ladder makes an angle of 60^o with the horizontal.
∴ ΔABC is a 30^o-60^o-90^o, right triangle.

Given: BC = 2.5 m, ∠ABC = 60°
AB = 5 cm and ∠BAC = 30°

From Pythagoras Theorem, we have
AB² = BC² + CA²
5² = (2.5)² + (CA)²
(CA)² = 25 – 6.25 = 18.75 m
Hence, length of the ladder is $square root of 18.75 end root space almost equal to 4.33 straight m$

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Short Answer Type

170. A ladder leaning against a wall makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.

170.
A ladder leaning against a wall makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.