The horizontal distance between two towers is 120 m. The angle of elevation of the top and angle of depression of the bottom of the first tower as observed from the second tower is 30° and 24° respectively. Find the heights of the two towers. Give your answer correct to 3 significant figures.

From figure,

AB is the first tower and CD is the second tower.

From figure,

AC = ED = 120 m.

In ΔBED,

$\Rightarrow \tan \theta = \dfrac{\text{perpendicular}}{\text{base}} \\[1em] \Rightarrow \tan 30^{\circ} = \dfrac{BE}{ED} \\[1em] \Rightarrow \dfrac{1}{\sqrt{3}} = \dfrac{BE}{120} \\[1em] \Rightarrow BE = \dfrac{120}{\sqrt{3}} \\[1em] \Rightarrow BE = \dfrac{120}{1.732} \\[1em] \Rightarrow BE = 69.3 \text{ m.}$

In ΔEDA,

$\Rightarrow \tan \theta = \dfrac{\text{perpendicular}}{\text{base}} \\[1em] \Rightarrow \tan 24^{\circ} = \dfrac{EA}{ED} \\[1em] \Rightarrow 0.445 = \dfrac{EA}{120} \\[1em] \Rightarrow EA = 120 \times 0.445 \\[1em] \Rightarrow EA = 53.4 \text{ m.}$

⇒ AB = AE + EB = 53.4 + 69.3 = 122.7 meters.

⇒ CD = EA = 53.4 meters.

Hence, height of two towers = 122.7 meters and 53.4 meters.