Question

A pole $10$m high cast a shadow$10$m long on the ground, then the sun’s elevation is:
A. $60^\circ $
B. $45^\circ $
C. $30^\circ $
D. $90^\circ $

Answer 1

Hint: Try to define the whole environment of the problem using the figure. Then apply the trigonometric ratio to get the desired result.

Complete step by step solution:
It is given that a pole $10$m high cast a shadow$10$m long on the ground.
The goal of the problem is to find the elevation of the sun, so that the length of the shadow is $10{\text{m}}$ on the ground.
First assume that the length of the pole as ${\text{AB}}$ and the length of the shadow of the pole is $AC$.
Now, make a figure that defines the given problem:

Assume that $\theta $ be the angle of depression that the sun makes. Then using the property of alternate angles, we have the angle $\angle C$ as $\theta $.
Now, apply the property of trigonometric ratios in the triangle $\Delta ABC$. Then the tangent of the angle $\theta $ is equal to the ratio of the opposite side and adjacent side. That is,
$\tan \theta = \dfrac{{{\text{Opposite side}}}}{{{\text{Adjacent side}}}}$
In the above figure, we can see that the opposite side is the length of the pole, whose value is $10$m and the adjacent side is the length of the shadow of the pole, whose value is $10$m. Substitute these values in the above relation:
$\tan \theta = \dfrac{{10}}{{10}}$
$ \Rightarrow \tan \theta = 1$
$ \Rightarrow \theta = {\tan ^{ - 1}}\left( 1 \right)$
We know that, $1$ can be expressed as:
$1 = \tan 45^\circ $
From the above relation, we have
$ \Rightarrow \theta = {\tan ^{ - 1}}\left( {\tan 45^\circ } \right)$
$ \Rightarrow \theta = 45^\circ $
So, the angle of elevation make by the sun is $45^\circ $.

Therefore, the option $\left( B \right)$ is correct.

Note:Using the given data, we had found the angle from the point C towards the top of the pole, which is not the required answer. We have to use the property of alternate angles to get the required angle that is made by the sun.