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An aeroplane, at an altitude of 250 m, observes the angle of depression of two boats on the opposite banks of a river to be 45° and 60° respectively. If the boats are on the opposite sides of the aeroplane, find the width of the river. Write the answer correct to the nearest whole number.

Heights & Distances

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Answer

Let D be the position of aeroplane and A and B be the positions of boat.

As, angle of depression are 60° and 45°.

Since, alternate angles are equal.

∴ ∠DAC = EDA = 60° and ∠DBC = FDB = 45°.

An aeroplane, at an altitude of 250 m, observes the angle of depression of two boats on the opposite banks of a river to be 45° and 60° respectively. If the boats are on the opposite sides of the aeroplane, find the width of the river. Write the answer correct to the nearest whole number. Heights and Distances, Concise Mathematics Solutions ICSE Class 10.

In △ACD,

tan 60°=PerpendicularBase⇒3=CDAC⇒AC=CD3⇒AC=2501.732⇒AC=144.34 m.\text{tan 60°} = \dfrac{\text{Perpendicular}}{\text{Base}} \\[1em] \Rightarrow \sqrt{3} = \dfrac{CD}{AC} \\[1em] \Rightarrow AC = \dfrac{CD}{\sqrt{3}} \\[1em] \Rightarrow AC = \dfrac{250}{1.732} \\[1em] \Rightarrow AC = 144.34 \text{ m}.

In △BCD,

tan 45°=PerpendicularBase⇒1=CDBC⇒BC=CD=250 m.\text{tan 45°} = \dfrac{\text{Perpendicular}}{\text{Base}} \\[1em] \Rightarrow 1 = \dfrac{CD}{BC} \\[1em] \Rightarrow BC = CD = 250 \text{ m}.

AB = AC + BC = 144.34 + 250 = 394.34 ≈ 394 m.

Hence, width of river = 394 m.

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