Height & Distance – Wire length from angle Two poles are 20 m and 14 m high. Their tops are connected by a straight wire that makes a 30° angle with the horizontal. What is the length of the wire?

Introduction / Context:
When a straight wire spans between unequal heights, its inclination with the horizontal lets us relate the vertical difference to the wire’s length via basic trigonometry. The scenario forms a right triangle whose “opposite” is the height difference and whose hypotenuse is the wire.

Given Data / Assumptions:

• Taller pole = 20 m; shorter pole = 14 m.
• Wire joins their tops; inclination to horizontal = 30°.
• Ground is level; wire is taut and straight.

Concept / Approach:
Let L be the wire length. Vertical rise between ends is Δh = 20 − 14 = 6 m. With angle θ to the horizontal, sin θ = (vertical rise)/L = Δh/L. Hence L = Δh / sin θ.

Step-by-Step Solution:

Verification / Alternative check:
Using components: vertical = L * sin 30° = L/2. Set L/2 = 6 ⇒ L = 12, consistent.

Why Other Options Are Wrong:
10 m and 8 m correspond to sin values inconsistent with 30° for a 6 m rise.

Common Pitfalls:
Accidentally using tan instead of sin (tan ties vertical to horizontal span, not to hypotenuse), or mixing which pole is taller.

Final Answer:
12 m