The tops of two poles of heights 20 m and 28 m are connected with a wire. The wire is inclined to the horizontal at an angle of $30°$. Find the length of the wire and the distance between the two poles.
Generated by claude-sonnet-4-6 · 2026-06-15 10:36 · grounding rag
Model Answer
Let the two poles AB = 20 m and CD = 28 m be standing vertically. The wire connects their tops, i.e., A and C.
Difference in heights = 28 − 20 = 8 m = CE (where E is the point on CD such that AE is horizontal).
In right △AEC, the wire AC is inclined at 30° to the horizontal.
Length of wire (AC):
$$\sin 30° = \frac{CE}{AC}$$
$$\frac{1}{2} = \frac{8}{AC}$$
$$AC = 16 \text{ m}$$
Distance between the poles (AE = BD):
$$\cos 30° = \frac{AE}{AC}$$
$$\frac{\sqrt{3}}{2} = \frac{AE}{16}$$
$$AE = 8\sqrt{3} \text{ m}$$
∴ The length of the wire is 16 m and the distance between the poles is $8\sqrt{3}$ m.
Source: Chapter 9 — Some Applications of Trigonometry
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Explanation
- The key step is finding the vertical difference between the two poles (28 − 20 = 8 m), which becomes the "opposite side" in the right triangle formed by the wire, the vertical height difference, and the horizontal distance.
- Use sin 30° for the wire length (hypotenuse) and cos 30° for the horizontal distance (distance between poles).
- Examiners expect a clear diagram description or labelling, correct trig ratio setup, and both answers stated explicitly.