Step 1: Let $AB$ be the tower and $BC$ the flagstaff. Let $P$ be the point on the ground with $AP = 30$ m.
The angle of elevation of $B$ (bottom of flagstaff $=$ top of tower) is $30^\circ$ and of $C$ (top of flagstaff) is $60^\circ$.

Height of the tower
Step 2: In right $\triangle ABP$,
$$\tan 30^\circ = \frac{AB}{30} \Rightarrow \frac{1}{\sqrt{3}} = \frac{AB}{30}$$
Step 3: $$AB = \frac{30}{\sqrt{3}} = 10\sqrt{3} = 10 \times 1{\cdot}73 = 17{\cdot}3\ \text{m}$$

Length of the wire attached to the bottom of the flagstaff ($PB$)
Step 4: $$\cos 30^\circ = \frac{30}{PB} \Rightarrow \frac{\sqrt{3}}{2} = \frac{30}{PB}$$
$$PB = \frac{60}{\sqrt{3}} = 20\sqrt{3} = 20 \times 1{\cdot}73 = 34{\cdot}6\ \text{m}$$

Length of the wire attached to the top of the flagstaff ($PC$)
Step 5: $$\cos 60^\circ = \frac{30}{PC} \Rightarrow \frac{1}{2} = \frac{30}{PC} \Rightarrow PC = 60\ \text{m}$$
Step 6: (Height of flagstaff, if needed : $AC = 30\tan 60^\circ = 30\sqrt{3} = 51{\cdot}9$ m, so $BC = 51{\cdot}9 - 17{\cdot}3 = 34{\cdot}6$ m.)

Hence height of the tower $= 17{\cdot}3$ m, and the wires are $34{\cdot}6$ m and $60$ m long.