Question

Two poles of equal heights are standing opposite 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 poles.

Hint:

For any equal height pole problem where angles of elevation are \(60^\circ\) and \(30^\circ\), the observation point always divides the total distance in a \(1 : 3\) ratio!
Since total width is 80 m, the segments are easily found as \(20\text{ m}\) and \(60\text{ m}\).

Answer 1

Correct Answer: 60

Step 1: Set up the variables.
Let each pole have height \(h\) metres. The road is 80 m wide. Let the observation point be at distance \(x\) m from the pole with elevation \(60^\circ\), so it is at \((80 - x)\) m from the pole with elevation \(30^\circ\).
Step 2: Form the first equation using \(60^\circ\).
\(\tan 60^\circ = \frac{h}{x}\), so \(\sqrt{3} = \frac{h}{x}\), giving \(h = \sqrt{3}x\).
Step 3: Form the second equation using \(30^\circ\).
\(\tan 30^\circ = \frac{h}{80 - x}\), so \(\frac{1}{\sqrt{3}} = \frac{h}{80-x}\), giving \(h = \frac{80-x}{\sqrt{3}}\).
Step 4: Equate the two expressions for h.
\(\sqrt{3}x = \frac{80-x}{\sqrt{3}}\). Multiply both sides by \(\sqrt{3}\): \(3x = 80 - x\). So \(4x = 80\), giving \(x = 20\) m.
Step 5: Find the height h.
\(h = \sqrt{3} \times 20 = 20\sqrt{3}\) m.
Step 6: State the complete answer.
The distance from the observation point to the nearer pole is \(x = 20\) m, and to the farther pole is \(80 - 20 = 60\) m.
\[ \boxed{h = 20\sqrt{3} \text{ m},\quad \text{distances: } 20 \text{ m and } 60 \text{ m}} \]