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The pilot of helicopter at an altitude of 1000 m sees two aeroplanes, one on his left and the other on his right at the same height and finds their angles of depression as $45 °$ and $60 °$ . What will be the distance between the two aeroplanes?

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1577.4m

1487.4m

1578.4m

1577m

answer is A.

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Detailed Solution

Given that,
The angle of depression of the left aeroplane will be

θ = 45 °

The angle of depression of the right aeroplane will be

ϕ = 60 °

The altitude of the helicopter from the ground will be

h = 1000

m We draw a diagram.
Question Image

Δ P R S

,
PS the distance from the helicopter to the right aeroplane let x.
SR= the altitude at which the helicopter stays from the ground
From

Δ P R S

using

tan ϕ = h e i g h t b a s e

tan ϕ = S R P S ⇒ tan ϕ = h x ⇒ x = h tan ϕ

Substituting

h

1000

m and

ϕ = 60 °

x = h tan ϕ

⇒ x = 1000 tan 60 ° ⇒ x = 10003 [∵ tan 60 ° = 3] ⇒ x = 577.35 m

Δ S Q R

,
QS = the distance from the helicopter to the left aeroplane let y.
From

Δ S Q R

tan θ = S R Q S ⇒ tan θ = h y ⇒ y = h tan θ

Substituting h=

1000

m and

θ

45 °

y = h tan θ

⇒ y = 1000 tan 45 ° ⇒ y = 10001 [∵ tan 45 ° = 1] ⇒ y = 1000 m

Using

x

577.35

m and y=

1000

m and

d

is the distance between the aeroplanes,

d = x + y ⇒ d = 1000 +577 .35 ⇒ d=1577 .4 m

Thus the distance between the aeroplanes will be 1577.4m.
Therefore the correct option is 1.

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The pilot of helicopter at an altitude of 1000 m sees two aeroplanes, one on his left and the other on his right at the same height and finds their angles of depression as 45 ° and 60° . What will be the distance between the two aeroplanes?

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The pilot of helicopter at an altitude of 1000 m sees two aeroplanes, one on his left and the other on his right at the same height and finds their angles of depression as $45 °$ and $60 °$ . What will be the distance between the two aeroplanes?