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 \[{{30}^{\circ }}\] and \[{{24}^{\circ }}\] respectively. Find the height of the two towers. Give your answer correct to 4 significant figures.

Hint: For the given question, we will consider the \[\Delta BCD\] from the given figure and we will use the tangent of \[\angle CBD\] which will give us the value of CD. Again, we will consider \[\Delta AEC\] and we will use the tangent of \[\angle ACE\] which will give us the value of AE.

Complete step-by-step answer:

We have been given the distance between two towers is 120 m and the angle of elevation of top and angle of depression of the bottom of first tower as observed from the second tower is \[{{30}^{\circ }}\] and \[{{24}^{\circ }}\] respective which are shown in the figure as follows:



Now in \[\Delta BCD\], we have \[\angle CBD={{24}^{\circ }}\] and BD = 120 m. Since it is a right angled triangle, we can find \[\tan \theta =\dfrac{perpendicular}{base}\].
\[\begin{align}
  & \tan {{24}^{\circ }}=\dfrac{CD}{BD} \\
 & \tan {{24}^{\circ }}=\dfrac{CD}{120} \\
 & CD=120\times \tan {{24}^{\circ }} \\
\end{align}\]
As we know that the approximate value of \[\tan {{24}^{\circ }}=0.4452\]
\[\begin{align}
  & CD=120\times 0.4452 \\
 & CD=53.42m \\
\end{align}\]
Once again, in \[\Delta AEC\] we have EC = BD = 120 m and \[\angle ACE={{30}^{\circ }}\].
Since it is a right angled triangle we can find \[\tan \theta =\dfrac{perpendicular}{base}\].
\[\begin{align}
  & \tan {{30}^{\circ }}=\dfrac{AE}{EC} \\
 & \tan {{30}^{\circ }}=\dfrac{AE}{120} \\
 & AE=120\times \tan {{30}^{\circ }} \\
\end{align}\]
As we know that the approximate value of \[\tan {{30}^{\circ }}=0.577\].
\[\begin{align}
  & AE=120\times 0.577 \\
 & AE=69.24m \\
\end{align}\]
Also, we know that AB = AE + BE and BE = CD.
\[AB=AE+CD=69.24+53.42=122.66m\]
We can also write it as \[122.7m\]
Therefore, the height of the towers are 53.42 m and 122.7 m approximately correct upto 4 significant figures.

Note: Be careful while substituting the values of tangent in the equation as there is a chance of multiplication error.
We can also solve the above question if we consider the other triangles from the figure.
Also, take care of the significant figures of your final answer.