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^{\circ} and 60^{\circ} respectively. Find the width of the river. Write the answer correct to the nearest whole number.

ICSE Class X Math 2014 Solved Paper - Question 52

Question : 52 of 52

Marks: +1, -0

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. Find the width of the river. Write the answer correct to the nearest whole number.

Solution:

Let

A D=250 {\m}

height of aeroplane
Two boats are at B and C.

Let

\;\; {\BD}=x

and

{\DC}=y

From

△ {\ADB}

;

\;\;\text " From "\; △ {\ADB} ;\;\;{x}/{250}\;=\cot 45^{∘}

\;{x}/{250}\;=1

⇒ x\;=250 {\m}

From

△ {\ADC} ; \;\; \;{y}/{250}=\cot 60^{∘}

\;{y}/{250}\;=\;{1}/{{√{3}}}

⇒ \;\; y\;=250 × \;{1}/{{√{3}}} {\m}

Width of river

{\BC}= {\BD}+ {\DC}

\;=x+y

\;=250+\;{250}/{{√{3}}}

\;=250 (1+\;{1}/{{√{3}}})

\;=250 (\;{{√{3}}+1}/{{√{3}}})

\;=250 (\;{1.732+1}/{1.732}) =250 (\;{2.732}/{1.732})

\;=250 × 1.577

\;=394.25 {\m}=394 {\m} .

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