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

Two poles of height 25 m and 16 m have their tops connected by a wire. The wire makes an angle of 30° with the vertical.

The height difference between the two poles is:

$25 - 16 = 9$ m

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The wire, the vertical height difference, and the horizontal distance between the poles form a right triangle.

In this triangle:

• The vertical side (height difference) = 9 m
• The horizontal side (distance between poles) = unknown
• The wire makes 30° with the vertical side

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Using the angle with the vertical side:

$\tan(30°) = \frac{\text{Horizontal distance}}{\text{Vertical distance}}$

$\tan(30°) = \frac{\text{Distance between poles}}{9}$

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Since $\tan(30°) = \frac{1}{\sqrt{3}}$:

$\frac{1}{\sqrt{3}} = \frac{\text{Distance}}{9}$

$\text{Distance} = \frac{9}{\sqrt{3}}$

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Rationalizing the denominator:

$\text{Distance} = \frac{9}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$

$\text{Distance} = \frac{9\sqrt{3}}{3}$

$\text{Distance} = 3\sqrt{3}$ m

Therefore, the distance between the two poles is $3\sqrt{3}$ m.