Question

The shadow of a flagpole is 30 meters. If the altitude of the sun is at ${{30}^{\circ }}$, then what is the height of the flagpole.
(a) $30\sqrt{3}$ m
(b) $\dfrac{30}{\sqrt{3}}$ m
(c) 15 m
(d) 10 m

Answer 1

To solve this problem we need to use the trigonometric identity of tangent according to which in a right angled triangle the tangent of the base angle will be equal to the ratio of height and the base of that triangle. In the given problem if we consider the height of the flag pole as the height of the triangle and base as shadow formed. Now by joining the end points of height and base we will get a right angled triangle and after applying the tangent function on the triangle we will get the height of the flag pole as answer.

Complete step by step answer:
To solve this problem we need to know that in the right angle triangle $\Delta ABC$,

The trigonometric function $\tan \theta$ is given by the ratio of AB to AC, i.e
$\tan \theta =\dfrac{AB}{AC}$
Now we are given the question that the shadow formed by the sun of the flagpole is 30 meters at ${{30}^{\circ }}$ angle of altitude.
Now we have to find the height(h) of the flagpole,
If in the above triangle, we assume AB as the height of the flagpole and BC as the shadow formed by the sun and $\theta ={{30}^{\circ }}$ then after we will apply the tangent function in the triangle to get the height i.e. AB, so we get

Now applying the tangent function in the triangle, we get
$\tan {{30}^{\circ }}=\dfrac{AB}{AC}$
$\tan {{30}^{\circ }}=\dfrac{AB}{AC}=\dfrac{h}{30}$
Now value of $\tan {{30}^{\circ }}$ is equal to $\dfrac{1}{\sqrt{3}}$, putting this in above equation we get
$\begin{aligned} & \dfrac{1}{\sqrt{3}}=\dfrac{h}{30} \\\ & h=\dfrac{30}{\sqrt{3}}=10\sqrt{3} \\\ \end{aligned}$
Hence the height of the flagpole is $\dfrac{30}{\sqrt{3}}$ hence we get option (b) as the correct answer.
Note:
Some students may make mistakes while applying the tangent function and may apply the reciprocal of AC and AB equal to tangent which is wrong and so you need to be careful while solving that. And also be careful while choosing the angle ${{30}^{\circ }}$ you may choose the angle $\angle BAC$ as ${{30}^{\circ }}$ which is also wrong.