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From an aeroplane, the angles of depression of two ships in a river on left and right of it are $60 °$ and $45 °$ respectively. If the distance between the two ships is 100m, then what will be the height of the aeroplane?

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64m

63m

63.4m

62.1m

answer is C.

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Detailed Solution

Given that, The distance between the two ships

P Q = 100

m and the angles of depression of two ships in a river on left and right of it are

60 °

and

45 °

.
We draw a schematic 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

By the tangent rule in

Δ R P S

tan 60 ° = R S P S ⇒ 3 = h x [∵ tan 60 ° = 3] ⇒ x = h 3

Applying the tangent rule in

Δ R S Q

tan 45 ° = R S S Q ⇒ 1 = h 100 − x [∵ tan 45 ° = 1] ⇒ h = 100 − x ⇒ h + x = 100

For the equation of the height h we substitute

x = h 3

h + x = 100

⇒ h + h 3 = 100 ⇒ 3 h + h 3 = 100 ⇒ 3 h + h = 1003

⇒ h = 173.2 − 300 − 2 ⇒ h = − 126.8 − 2 ⇒ h = 63.4 m

Thus, the height of the aeroplane will be 63.4 m.
Therefore the correct option is 3.

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