A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff. From a point on the ground $\displaystyle 30$ m away from the tower, wires are attached to the top and bottom of the flagstaff making angles of elevation $\displaystyle 60^{\circ}$ and $\displaystyle 30^{\circ}$ respectively. Find the height of the tower and lengths of the wires attached. (Take $\displaystyle \sqrt{3}=1 \cdot 73$ )

Let AB be the tower of height 'h' m and BC be the flagstaff. In right angled \(\displaystyle \Delta \mathrm{BAD}\) \[\frac{\mathrm{h}}{30}=\tan 30^{\circ}=\frac{1}{\sqrt{3}} \] \(\displaystyle \Rightarrow \mathrm{h}=\frac{30}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}=10 \sqrt{3}=17.3\) ∴ height of the tower is $\displaystyle 17.3$ m Also, \(\displaystyle \frac{30}{l_{1}}=\cos 30^{\circ}=\frac{\sqrt{3}}{2}\) \(\displaystyle \Rightarrow l_{1}=\frac{60}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}=20 \sqrt{3}=34.6\) ∴ length of wire attached to the bottom of the flagstaff is $\displaystyle 34.6$ m In right angled \(\displaystyle \triangle \mathrm{CAD}\) \[\frac{30}{l_{2}}=\cos 60^{\circ}=\frac{1}{2} \] \(\displaystyle \Rightarrow l_{2}=60\) ∴ length of wire attached to the top of the flagstaff is $\displaystyle 60$ m