Question

A vertical pole fixed to the horizontal ground is divided in the ratio $3:7$ by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground $18$ m away from the base of the pole, then the height of the pole (in meters) is:

• A. $8\sqrt{10}$
• B. $12\sqrt{10}$
• C. $12\sqrt{15}$
• D. $6\sqrt{10}$

Answer 1

The correct option is B $12\sqrt{10}$

$\tan \theta =\frac{3h}{18}=\frac{h}{6}\tan 2\theta =\frac{10h}{18}=\frac{5h}{9}\Rightarrow \frac{2\tan \theta }{1-{\tan }^2\theta }=\frac{5h}{9}\Rightarrow \frac{2\left(\frac{h}{6}\right)}{1-\frac{h^2}{36}}=\frac{5h}{9}\Rightarrow \frac{12}{36-h^2}=\frac{5}{9}\Rightarrow 108=180-5h^2\Rightarrow h^2=\frac{72}{5}\Rightarrow h=6\sqrt{\frac{2}{5}}[\because h>0]\therefore 10h=60\sqrt{\frac{2}{5}}=12\sqrt{10}$
Hence, the height of the pole is $12\sqrt{10}$ meters