The tops of two poles of height 38 m and 56 m are connected by a cable. If the cable makes an angle of 30° with the horizontal, then the distance between the bases of the poles is:

$18\sqrt{3}m$

The correct answer is Option (2) → $18\sqrt{3}m$

Step-by-Step Calculation:

To solve this, we can visualize the two poles as vertical lines and the cable as the hypotenuse of a right-angled triangle formed at the top.

1. Find the difference in height:

The two poles have heights of 56 m and 38 m. The vertical side of the triangle formed by the cable ($h$) is the difference between these two heights:

$h = 56\text{ m} - 38\text{ m} = 18\text{ m}$

2. Identify the Triangle:

Let the distance between the bases of the poles be $d$. This distance is the same as the horizontal base of the triangle formed at the top.

• The angle of elevation ($\theta$) = 30°
• The opposite side (height difference) = 18 m
• The adjacent side (distance between poles) = $d$

3. Use the Trigonometric Ratio:

We use the tangent function ($\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$):

$\tan 30^\circ = \frac{18}{d}$

We know that $\tan 30^\circ = \frac{1}{\sqrt{3}}$, so:

$\frac{1}{\sqrt{3}} = \frac{18}{d}$

4. Solve for $d$:

$d = 18 \times \sqrt{3}$

$d = \mathbf{18\sqrt{3} \text{ m}}$