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Mathematics · 2023 · 5 marks
CBSE 2023 · Region 4 · Set 1 · Q32
(A) A straight highway leads to the foot of a tower. A man standing on the top of the 75 m high tower observes two cars at angles of depression of…
A straight highway leads to the foot of a tower. A man standing on the top of the $\displaystyle 75$ m high tower observes two cars at angles of depression of $\displaystyle 30^{\circ}$ and $\displaystyle 60^{\circ}$, which are approaching the foot of the tower. If one car is exactly behind the other on the same side of the tower, find the distance between the two cars. (use $\displaystyle \sqrt{3}=1.73$ )
From the top of a $\displaystyle 7$ m high building, the angle of elevation of the top of a cable tower is $\displaystyle 60^{\circ}$ and the angle of depression of its foot is $\displaystyle 30^{\circ}$. Determine the height of the tower.
Marking-scheme solution
$\displaystyle \mathrm{AB}=$ Height of tower $\displaystyle =75 \mathrm{~m}$
P, Q are positions of cars
\[\begin{array}{l}
\angle \mathrm{XBQ}=\angle \mathrm{BQA}=30^{\circ} \\
\angle \mathrm{XBP}=\angle \mathrm{BPA}=60^{\circ}
\end{array}
\]
In $\displaystyle \Delta \mathrm{APB}, \tan 60^{\circ}=\frac{75}{\mathrm{AP}} \Rightarrow \mathrm{AP}=\frac{75}{\sqrt{3}}=25 \sqrt{3}$
In $\displaystyle \Delta \mathrm{AQB}, \tan 30^{\circ}=\frac{75}{\mathrm{AQ}} \Rightarrow \mathrm{AQ}=75 \sqrt{3}$
Distance between the cars $\displaystyle =\mathrm{PQ}=\mathrm{AQ}-\mathrm{AP}$
\[\begin{array}{l}
=75 \sqrt{3}-25 \sqrt{3}=50 \sqrt{3} \\
=50 \times 1 \cdot 73=86 \cdot 5 \mathrm{~m}
\end{array}
\]
Let AC be $\displaystyle \mathrm{h} \mathrm{m}, \mathrm{BC}=\mathrm{DE}=7 \mathrm{~m}, \mathrm{AB}=(\mathrm{h}-7) \mathrm{m}$
$\displaystyle \therefore \angle E C D=30^{\circ}$
Let CD be $\displaystyle x \mathrm{~m}$
$\displaystyle \frac{D E}{C D}=\frac{7}{x}=\tan 30^{\circ} \Rightarrow x=7 \sqrt{3}$
$\displaystyle \Rightarrow B E=7 \sqrt{3}$
Again $\displaystyle \frac{A B}{B E}=\tan 60^{\circ}$
$\displaystyle \Rightarrow \frac{h-7}{7 \sqrt{3}}=\sqrt{3}$
$\displaystyle \Rightarrow h=28$
∴ Height of tower $\displaystyle =28 m$
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