Question

Tower of height $50$ metres stands on a level ground. A flag-staff standing on the tower of length $hm$ subtends an angle of ${\tan }^{-1}\frac{1}{3}$ at a point $100$ metres away from the tower on the ground. The length of the flag-staff in metres is:

• A. $50$
• B. $75$
• C. $100$
• D. $125$

Answer 1

The correct option is C $100$

In $△PQT,\tan \alpha =\frac{150-h}{x}\Rightarrow \frac{4}{3}=\frac{150-h}{x}$

$\Rightarrow 4x=450-3h\Rightarrow h=\frac{1}{3}(450-4x)\to 1$

$△PRS,\tan \beta =\frac{150}{x}\Rightarrow \frac{5}{2}=\frac{150}{x}$

$\Rightarrow x=60m$

$\Rightarrow h=\frac{1}{3}(450-240)=70$

In $△PQT$

$PQ=150-70=80m$

$QT=60m$

PT=distance between top$=\sqrt{{PQ}^2+{QT}^2}$

$=\sqrt{{\left(80\right)}^2+{60}^2}=\sqrt{6400+3600}=\sqrt{10000}=100m$