Question

Two poles of equal heights are standing opposite each other on either side of the road which is 85 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 distances of the point from the poles. (Use \(\sqrt{3} = 1.73\))

Answer 1

Given:
- Two poles of equal height \(h\) on opposite sides of a road 85 m wide.
- Point \(P\) lies between the poles on the road.
- Angles of elevation from \(P\) to the top of the poles are \(60^\circ\) and \(30^\circ\).
- Distance between the poles = 85 m.
- Use \(\sqrt{3} = 1.73\).

Step 1: Let the distances of point \(P\) from the poles be \(x\) and \(85 - x\) meters.
Step 2: Use tangent of angles for each pole
For the pole with angle of elevation \(60^\circ\):
\[ \tan 60^\circ = \frac{h}{x} \implies \sqrt{3} = \frac{h}{x} \implies h = \sqrt{3} \, x = 1.73 \, x \]

For the pole with angle of elevation \(30^\circ\):
\[ \tan 30^\circ = \frac{h}{85 - x} \implies \frac{1}{\sqrt{3}} = \frac{h}{85 - x} \implies h = \frac{85 - x}{\sqrt{3}} = \frac{85 - x}{1.73} \]

Step 3: Equate the two expressions for height \(h\)
\[ 1.73 \, x = \frac{85 - x}{1.73} \] Multiply both sides by 1.73:
\[ 1.73 \times 1.73 \, x = 85 - x \] \[ 3 \, x = 85 - x \] \[ 3x + x = 85 \] \[ 4x = 85 \implies x = \frac{85}{4} = 21.25 \, \text{m} \]

Step 4: Calculate height \(h\)
\[ h = 1.73 \times 21.25 = 36.76 \, \text{m} \]

Step 5: Calculate distance from other pole
\[ 85 - x = 85 - 21.25 = 63.75 \, \text{m} \]

Final Answer:
- Height of the poles: \(\boxed{36.76 \, \text{m}}\)
- Distance of the point from the first pole: \(\boxed{21.25 \, \text{m}}\)
- Distance of the point from the second pole: \(\boxed{63.75 \, \text{m}}\)