The tops of two poles of height 25 m and 16 m are connected by a wire. If the wire makes an angle 30° with the vertical, then the distance between the two poles is

$3\sqrt{3}m$

The correct answer is Option (4) → $3\sqrt{3}m$

1. Identify the given information

• Height of Pole 1 ($H_1$): $25\text{ m}$
• Height of Pole 2 ($H_2$): $16\text{ m}$
• Difference in height ($h$): The vertical distance between the tops of the two poles is $25\text{ m} - 16\text{ m} = 9\text{ m}$.
• Angle with the vertical ($\theta$): $30^\circ$.

2. Set up the geometry

Imagine a horizontal line drawn from the top of the shorter pole ($16\text{ m}$) to the taller pole ($25\text{ m}$). This creates a right-angled triangle where:

• The vertical side (adjacent to the angle with the vertical) is the difference in heights: $9\text{ m}$.
• The horizontal side (opposite to the angle with the vertical) is the distance between the poles, let's call it $d$.
• The angle between the wire (hypotenuse) and the vertical side is $30^\circ$.

3. Calculate the distance ($d$)

Using the trigonometric ratio for tangent ($\tan$):

$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

$\tan(30^\circ) = \frac{d}{9}$

We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}}$. Substituting this value into the equation:

$\frac{1}{\sqrt{3}} = \frac{d}{9}$

$d = \frac{9}{\sqrt{3}}$

To simplify, multiply the numerator and denominator by $\sqrt{3}$:

$d = \frac{9\sqrt{3}}{3} = 3\sqrt{3}\text{ m}$

Final Answer:

The distance between the two poles is $3\sqrt{3}\text{ m}$.