Question

Two poles of equal heights are standing opposite to each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are $60^\circ$ and $30^\circ$ respectively. Find the height of the poles and the distance of the point from the pole.

Hint:

When two angles of elevation are given from a point, use $\tan \theta = \dfrac{\text{height}}{\text{distance}}$ for both and solve the equations simultaneously.

Answer 1

Step 1: Let the height of each pole be $h$ m. Let the distance of the point from the pole at $60^\circ$ be $x$ m, then the distance from the other pole is $(80 - x)$ m.

Step 2: Apply trigonometric ratios.
For the first pole, \[ \tan 60^\circ = \dfrac{h}{x} \Rightarrow \sqrt{3} = \dfrac{h}{x} \Rightarrow h = \sqrt{3}x \] For the second pole, \[ \tan 30^\circ = \dfrac{h}{80 - x} \Rightarrow \dfrac{1}{\sqrt{3}} = \dfrac{h}{80 - x} \Rightarrow h = \dfrac{80 - x}{\sqrt{3}} \]
Step 3: Equate both values of $h$.
\[ \sqrt{3}x = \dfrac{80 - x}{\sqrt{3}} \Rightarrow 3x = 80 - x \Rightarrow 4x = 80 \Rightarrow x = 20 \]
Step 4: Find the height.
\[ h = \sqrt{3}x = \sqrt{3} \times 20 = 20 \times 1.732 = 34.64 \, \text{m} \] Step 5: Conclusion.
Hence, the height of each pole is 34.6 m, and the distances from the point are 20 m and 60 m respectively.