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The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of elevation of the top of the tower at the foot of the hill is $30 °$ If the tower is 50 m tall, what is the height of the hill?

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150m

160m

140m

165m

answer is A.

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Detailed Solution

Given that,
Angle of Elevation top of the hill at the foot of the tower=

60 °

Angle of Elevation top of the tower at the foot of a hill=

30 °

The height of the tower =50m.
We draw the diagram,
Question Image

We use the trigonometric identity,

tan θ = p e r p e n d i c u l a r b a s e cot θ = b a s e p e r p e n d i c u l a r

We get the value of AC from

Δ B A C

⇒ cot 30 ° = A C A B ⇒ 3 = A C 50 [∵ cot 30 ° = 3] ⇒ A C = 503 m

Taking CD=x.
Then, we get the value of CD from

Δ A C D

⇒ tan 60 ° = C D A C ⇒ 3 = x 503 [∵ tan 60 ° = 3] ⇒ x = 503 × 3

⇒ x = 50 × 3 ⇒ x = 150 m

Thus the height of the hill is 150m.
Therefore the correct option is 1.

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