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Mathematics · 2022 · 3 marks
CBSE 2022 · Region 3 · Set 3 · Q7
Two boats are sailing in the sea 80 m apart from each other towards a cliff AB. The angles of depression of the boats from the top of the cliff are…
Two boats are sailing in the sea $\displaystyle 80$ m apart from each other towards a cliff AB. The angles of depression of the boats from the top of the cliff are $\displaystyle 30$° and $\displaystyle 45^{\circ}$ respectively, as shown in Figure 3. Find the height of the cliff.
Figure $\displaystyle 3$
Figure $\displaystyle 3$Marking-scheme solution
\[\begin{array}{l}
\tan 45^{\circ}=\frac{h}{x}=1 \\
\quad \Rightarrow h=x \ldots \ldots \ldots \text { (i) } \\
\tan 30^{\circ}=\frac{1}{\sqrt{3}}=\frac{h}{x+80} \ldots \ldots \ldots
\end{array}
\]Using (i) and (ii)
\[h=\frac{80}{\sqrt{3}-1}=40(\sqrt{3}+1) \mathrm{m}
\]
∴ Height of the cliff $\displaystyle =40(\sqrt{ } 3+1) \mathrm{m}$
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