Question

The length of the shadow of a vertical pole 9 m high, when the sun’s altitude is $30^\circ$, is ( in cm ) :
a)${\text{3}}\sqrt {\text{3}}$
b)$9$
c)$9\sqrt 3$
d)$18\sqrt 3$

Answer 1

In this question we use the ${{tan\theta = }}\dfrac{{{\text{perpendicular}}}}{{{\text{base}}}}$, where $\theta$ be the angle between the base and the hypotenuse of a right angle triangle and $\tan 30^\circ = \dfrac{1}{{\sqrt 3 }}$.

Complete step-by-step answer:

Let, AB be the vertical pole whose height is 9 m and BC be the length of the shadow, which is required.
Here, it is given that the sun’s altitude is $30^\circ$.
Now, we know that ${{tan\theta = }}\dfrac{{{\text{perpendicular}}}}{{{\text{base}}}}$, where $\theta$ be the angle between the base and the hypotenuse of a right angle triangle.
$\therefore$Therefore, from the triangle $\vartriangle$ABC, we write

$$\Rightarrow \tan 30^\circ = \dfrac{9}{{BC}}\left[ {\because \theta = 30^\circ } \right] \\\ \Rightarrow BC = \dfrac{9}{{\tan 30^\circ }} \\\ \Rightarrow BC = 9\cot 30^\circ = 9\sqrt 3 \left[ \begin{gathered} \because \tan 30^\circ = \dfrac{1}{{\sqrt 3 }} \\\ \therefore \cot 30^\circ = \dfrac{1}{{\tan 30^\circ }} = \sqrt 3 \\\ \end{gathered} \right] \\\$$

So, the length of the shadow is $9\sqrt 3 {\text{ }}m$.
Note: Here, you have to draw a clear diagram at the time of solving this type of height & distance problems.
After that, you have to know the properties of triangles and the trigonometric ratios.