Question

The angle of depression of a car, standing on the ground, from the top of 75m high tower, is \[{{30}^{\circ }}\] then the distance of car from the base of tower (in m) is:
(a) \[25\sqrt{3}\]
(b) \[50\sqrt{3}\]
(c) \[75\sqrt{3}\]
(d) \[100\]

Answer 1

Hint: We solve this problem first by drawing the rough figure of the given data as follows

We solve this problem by using the trigonometric ratio suitable to get the distance ‘BC’. Here, we use the tangent trigonometric ratio which is given as
\[\Rightarrow \tan \theta =\dfrac{\text{opposite side}}{\text{adjacent side}}\]

Complete step-by-step solution:
We are given that the height of tower as 75m
So, from the figure we can write
\[\Rightarrow AB=75m\]
We are given that the angle of depression of car as \[{{30}^{0}}\]
So, from the figure we can write
\[\Rightarrow \angle SAC={{30}^{\circ }}\]
Let us assume that
\[\Rightarrow \angle CAB=\theta \]
Now, from the figure we can divide the angle \[\angle A\] as follows
\[\Rightarrow \angle SAC+\angle CAB={{90}^{\circ }}\]
Now by substituting the required values in the above equation we get
\[\begin{align}
  & \Rightarrow {{30}^{\circ }}+\theta ={{90}^{\circ }} \\
 & \Rightarrow \theta ={{60}^{\circ }} \\
\end{align}\]
Now let us consider the triangle \[\Delta ABC\]
We know that the formula of tangent trigonometric ratio is given as
\[\Rightarrow \tan \theta =\dfrac{\text{opposite side}}{\text{adjacent side}}\]
By substituting the required values from \[\Delta ABC\] to above equation we get
\[\Rightarrow \tan {{60}^{\circ }}=\dfrac{BC}{AB}........equation(i)\]
We know that from the standard trigonometry table
\[\Rightarrow \tan {{60}^{\circ }}=\sqrt{3}\]
By substituting the values in equation (i) we get
\[\begin{align}
  & \Rightarrow \sqrt{3}=\dfrac{BC}{75m} \\
 & \Rightarrow BC=75\sqrt{3}m \\
\end{align}\]
Therefore the distance of car from the foot of tower is \[75\sqrt{3}\] meters. So, option (c) is the correct answer.

Note: Students may make mistakes in taking the angle of depression. The angle of depression or elevation always measures from the horizontal.
So, the figure will be

But students may misunderstand this and take the figure as

This gives the wrong answer. This point needs to be taken care of, that is the definition of the angle of depression.