Search any question & find its solution
Question:
Answered & Verified by Expert
If the angles of depression of the top and bottom of a short building from the top of a tall building are $30^{\circ}$ and $60^{\circ}$ respectively, then the ratio of the heights of short and tall buildings is
Options:
Solution:
2956 Upvotes
Verified Answer
The correct answer is:
$2: 3$
Let, $A B$ is tall building $C Q$ is short building
$\begin{aligned}
& A B=H \\
& C Q=h
\end{aligned}$
In $\triangle A B C$
$\begin{aligned}
\tan 60^{\circ} & =\frac{H}{A C} \\
A C & =\frac{H}{\sqrt{3}}
\end{aligned}$
$\begin{aligned}
& \text {In } \triangle B P Q \\
& \tan 30^{\circ}=\frac{H-h}{P Q} \\
& \Rightarrow \quad P Q=(H-h) \sqrt{3} \\
& A C=P Q \\
& \therefore \quad \frac{H}{\sqrt{3}}=(H-h) \sqrt{3}
\end{aligned}$
$\begin{aligned}
\Rightarrow & H =3 H-3 h \\
\Rightarrow & 2 H =3 h \\
\Rightarrow & \frac{h}{H} =\frac{2}{3} \\
\therefore & h: H =2: 3
\end{aligned}$
$\begin{aligned}
& A B=H \\
& C Q=h
\end{aligned}$
In $\triangle A B C$
$\begin{aligned}
\tan 60^{\circ} & =\frac{H}{A C} \\
A C & =\frac{H}{\sqrt{3}}
\end{aligned}$
$\begin{aligned}
& \text {In } \triangle B P Q \\
& \tan 30^{\circ}=\frac{H-h}{P Q} \\
& \Rightarrow \quad P Q=(H-h) \sqrt{3} \\
& A C=P Q \\
& \therefore \quad \frac{H}{\sqrt{3}}=(H-h) \sqrt{3}
\end{aligned}$
$\begin{aligned}
\Rightarrow & H =3 H-3 h \\
\Rightarrow & 2 H =3 h \\
\Rightarrow & \frac{h}{H} =\frac{2}{3} \\
\therefore & h: H =2: 3
\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.