Question

The tops of two poles of heights \(22\text{ m}\) and \(31\text{ m}\) are connected by a wire. If the wire makes an angle of \(60^\circ\) with the horizontal, then find the length of the wire.

• A. \(6\sqrt{3}\)
• B. \(3\sqrt{3}\)
• C. \(\dfrac{6}{\sqrt{3}}\)
• D. \(\dfrac{3}{\sqrt{3}}\)

Hint:

Whenever a wire, ladder, or pole makes an angle with the horizontal: \[ \sin\theta = \frac{\text{Vertical Height}}{\text{Length}} \] Use right triangle trigonometry carefully.

Answer 1

The Correct Option is A

Concept: When two poles are connected by a wire, a right triangle is formed. Trigonometric ratios can then be applied.

Step 1: Find the vertical difference between the poles. Heights are: \[ 31\text{ m and }22\text{ m} \] Difference: \[ 31 - 22 = 9\text{ m} \] This becomes the vertical side of the right triangle.

Step 2: Understand the geometry. The wire makes an angle of: \[ 60^\circ \] with the horizontal. Let the wire length be \(L\). Using sine ratio: \[ \sin 60^\circ = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \] \[ \frac{\sqrt{3}}{2} = \frac{9}{L} \]

Step 3: Solve for \(L\). \[ L = \frac{9 \times 2}{\sqrt{3}} \] \[ = \frac{18}{\sqrt{3}} \] Rationalizing: \[ = \frac{18\sqrt{3}}{3} \] \[ = 6\sqrt{3} \] Hence, \[ \boxed{6\sqrt{3}} \]