To determine the length of the wire connecting the tops of the poles, we can use trigonometry. Here's the setup:

• The height difference between the poles is $h = 40 - 30 = 10 \, \text{m}$.
• The wire makes an angle of $30^\circ$ with the horizontal.

Step-by-Step Solution:

1. Identify the Relevant Triangle: The wire forms the hypotenuse of a right triangle, where:

   • The vertical side (opposite side) is $h = 10 \, \text{m}$.
   • The angle between the hypotenuse (wire) and the horizontal is $\theta = 30^\circ$.

2. Trigonometric Relation: Using the sine function: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ Substituting the values: $\sin(30^\circ) = \frac{10}{\text{Length of the wire}}$ Since $\sin(30^\circ) = 0.5$: $0.5 = \frac{10}{\text{Length of the wire}}$

3. Solve for the Length of the Wire: Rearrange to find the hypotenuse (length of the wire): $\text{Length of the wire} = \frac{10}{0.5} = 20 \, \text{m}.$

Final Answer:

The length of the wire is 20 meters.

Would you like more details or assistance with a similar problem? Here are some related questions for further exploration:

1. How do you determine the angle if the length of the wire and height difference are given?
2. What would be the length of the wire if the angle changes to $45^\circ$?
3. How does the length of the wire vary with the angle of inclination?
4. Can this problem be solved using cosine or tangent functions?
5. What if the horizontal distance between the poles was known—how would that affect the calculation?

Tip: Always verify the angle’s trigonometric value to ensure accurate calculations.