Question

From the top of a pillar of height $20m$, the angles of elevation and depression of the top and bottom of another pillar are ${30}^0$ and ${45}^0$ respectively. The height of the second pillar (in meters) is:

• A. $\frac{20(\sqrt{3}+1)}{\sqrt{3}}$
• B. $10$
• C. $10\sqrt{3}$
• D. $\frac{10}{\sqrt{3}}(\sqrt{3}+1)$

Answer 1

The correct option is A $\frac{20(\sqrt{3}+1)}{\sqrt{3}}$

REF Image:

$CB$ is the height of the second tower. $PA$ is the height of the first tower and $ABQP$ is the rectangle.

$BC=CQ+QBPQ=AB(\because \square PABQ),\angle CPQ=30°\tan 45°=\frac{PA}{AB}\Rightarrow AB=\frac{PA}{\tan 45°}=20m=PQ\therefore In\Delta PQC,\angle CPQ=30°\therefore \tan 30°=\frac{CQ}{PQ}\Rightarrow CQ=20\tan 30°\Rightarrow CQ=\frac{20}{\sqrt{3}}mIn\Delta PQB,\angle PBQ=45°\tan 45°=\frac{PQ}{QB}\Rightarrow QB=PQ=20m\therefore CB=CQ+QB=20+\frac{20}{\sqrt{3}}=\frac{20(\sqrt{3}+1)}{\sqrt{3}}m$