Question

A circus artist is climbing a 20m long rope, which is tightly stretched and tied from the top of a vertical pole of the ground. Find the height of the pole if the angle made by the rope with the ground level is ${{30}^{\circ }}$.

A. $12$
B. $10$
C. $8$
D. $6$

Answer 1

Hint: So, the figure $\Delta ABC$ is a right-angled triangle. Here we will use the trigonometrical relation of $\sin $. The triangle is a right-angled triangle.
AC being the hypotenuse. AB acts as the height in the figure which is the pole. With respect to $\angle BCA$ the sides, AB and AC will be in the relation of $\sin $.

Complete step-by-step solution:
We know that with respect to any angle $\alpha $ the trigonometrical value of $\sin \alpha $ represents
$\sin \alpha =\dfrac{height}{hypotenuse}$.
Let’s assume the height of the pole AB is $x$ m.
So, $AB=x$.
We find the trigonometrical relation between AB and AC with respect to $\angle BCA$.
So, $\sin 30=\dfrac{height}{hypotenuse}=\dfrac{AB}{AC}$.
Now, we place all the values and get $\dfrac{AB}{AC}=\dfrac{x}{20}=\sin 30$.
Solving the equation, we get the value of $x$.
$\begin{align}
  & \dfrac{x}{20}=\sin 30=\dfrac{1}{2} \\
 & \Rightarrow x=\dfrac{20}{2}=10 \\
\end{align}$
So, the height of the pole is 10m. The correct option is (B).

Note: At the time of solving the problem we never used the base which BC. The reason being we don’t need to find out its length as it’s the distance of the point of the rope fixed in the ground from the pole’s bottom part. Also, it’s connected with the length of the pole through the trigonometrical relation of $\tan \alpha =\dfrac{height}{base}$. We don’t have the length of the base. So, to get to the length of the pole using the base, we first need to find the value of the base using the hypotenuse which is obviously an unnecessary step.