The tops of two poles of height 22 m and 31 m are connected by a wire. If the wire makes an angle of 60° with the horizontal, then the length of the wire (in m) is:

The two poles have heights of 22 m and 31 m respectively. A wire connects their tops at an angle of 60° with the horizontal.

The height difference between the poles is:

$31 - 22 = 9$ m

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When viewed from the side, the wire forms a right-angled triangle where:

• The vertical side (height difference) = 9 m
• The hypotenuse = length of the wire
• The angle with horizontal = 60°

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The wire makes a 60° angle with the horizontal, and the vertical height difference (opposite side) is 9 m.

Using the sine ratio:

$\sin(60°) = \dfrac{\text{opposite}}{\text{hypotenuse}}$

$\sin(60°) = \dfrac{9}{\text{wire length}}$

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Since $\sin(60°) = \dfrac{\sqrt{3}}{2}$:

$\dfrac{\sqrt{3}}{2} = \dfrac{9}{\text{wire length}}$

$\text{wire length} \times \sqrt{3} = 18$

$\text{wire length} = \dfrac{18}{\sqrt{3}}$

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Rationalizing the denominator:

$\text{wire length} = \dfrac{18}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}}$

$\text{wire length} = \dfrac{18\sqrt{3}}{3}$

$\text{wire length} = 6\sqrt{3}$ m

Therefore, the length of the wire is $6\sqrt{3}$ m.