Mathematics

The angles of depression of two ships A and B on opposite sides of a light house of height 100 m are respectively 42° and 54°. The line joining the two ships passes through the foot of the light house.
(a) Find the distance between the two ships A and B.
(b) Give your final answer correct to the nearest whole number.
(Use mathematical tables for this question)

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Let ∠BCP = α and ∠ACP = β

From figure,

⇒ α + 54° = 90°

⇒ α = 90° - 54° = 36°.

⇒ β + 42° = 90°

⇒ β = 90° - 42° = 48°.

⇒ tan α = $\dfrac{BP}{CP}$

⇒ tan 36° = $\dfrac{BP}{100}$

⇒ 0.7265 = $\dfrac{BP}{100}$

⇒ BP = 0.7265 × 100 = 72.65 m

⇒ tan β = $\dfrac{AP}{CP}$

⇒ tan 48° = $\dfrac{AP}{100}$

⇒ 1.1106 = $\dfrac{AP}{100}$

⇒ AP = 1.1106 × 100 = 111.06 m

(a) From figure,

AB = AP + BP = 72.65 + 111.06 = 183.71 m

Hence, the distance between two ships = 183.71 m.

(b) On rounding off,

AB = 184 m.

Hence, the distance between two ships = 184 m.

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