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Mathematics · 2022 · 3 marks
CBSE 2022 · Region 1 · Set 1 · Q9
(a) The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the…
(a)
The angle of elevation of the top of a building from the foot of the tower is $\displaystyle 30$° and the angle of elevation of the top of the tower from the foot of the building is $\displaystyle 60$°. If the tower is $\displaystyle 50$ m high, then find the height of the building.
OR
(b)
From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are $\displaystyle 30^{\circ}$ and $\displaystyle 45^{\circ}$ respectively. If the bridge is at a height of $\displaystyle 3$ m from the banks, then find the width of the river.
Marking-scheme solution
$\displaystyle \tan 30^{\circ}=\frac{A B}{A X}$ and $\displaystyle \tan 60^{\circ}=\frac{50}{A X}$
\[\Rightarrow A B=\frac{1}{\sqrt{3}} A X \text { and } A X=\frac{50}{\sqrt{3}}
\]
Tower ($\displaystyle 50$ m)
\[\therefore A B=\frac{1}{\sqrt{3}} \cdot \frac{50}{\sqrt{3}}=\frac{50}{3} \mathrm{~m}
\]
across a river, the angles of depression of sides of the river are $\displaystyle 30^{\circ}$ and $\displaystyle 45^{\circ}$ ge is at a height of $\displaystyle 3$ m from the banks, e river.
$\displaystyle 5^{\circ}$
$\displaystyle 5^{\circ}$
$\displaystyle 45^{\circ}$
Here, $\displaystyle \frac{P D}{A D}=\tan 30^{\circ}=\frac{1}{\sqrt{3}} \Rightarrow A D=3 \sqrt{3} \mathrm{~m}$ and $\displaystyle \frac{P D}{B D}=\tan 45^{\circ}=1 \Rightarrow B D=3 \mathrm{~m}$
So, $\displaystyle \mathrm{AB}=A D+B D=(3 \sqrt{3}+3) \mathrm{m}=3(\sqrt{3}+1) \mathrm{m}$
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