The problem describes a vertical transmission tower and a wire that stabilizes it. Here is the restatement of the problem:

Problem:

A vertical transmission tower has a height of 83 m. The wire makes an angle of 64° with the level ground and is attached 20 m below the top of the tower. How long must the wire be to stabilize the tower?

Solution Outline:

1. Height of the wire attachment:
   The wire is attached 20 meters below the top of the tower, so the attachment height from the ground is: $h = 83 \, \text{m} - 20 \, \text{m} = 63 \, \text{m}$

2. Right triangle setup:
   The height from the ground to where the wire is attached (63 m) forms one side of a right triangle. The wire forms the hypotenuse, and the ground forms the adjacent side. The angle between the wire and the ground is 64°.

3. Using trigonometry:
   We can use the sine function to find the length of the hypotenuse (the wire). The sine of an angle in a right triangle is defined as: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ where:

   • $\theta = 64^\circ$
   • opposite = 63 m (height of wire attachment)
   • hypotenuse = length of the wire ($L$)

   Thus, we have: $\sin(64^\circ) = \frac{63}{L}$ Solving for $L$: $L = \frac{63}{\sin(64^\circ)}$ Using a calculator: $\sin(64^\circ) \approx 0.8988$ So, $L \approx \frac{63}{0.8988} \approx 70.09 \, \text{m}$

Final Answer:

The wire must be approximately 70.09 meters long.

Would you like a more detailed explanation on any part of the solution? Here are five related questions:

1. How is trigonometry used to solve problems involving right triangles?
2. What other trigonometric functions could be used to solve similar problems?
3. How does the angle between the wire and the ground affect the length of the wire?
4. What would happen to the wire length if the attachment point was changed?
5. Can we calculate the horizontal distance the wire covers along the ground?

Tip: Always ensure you’re using the correct trigonometric ratio (sine, cosine, or tangent) based on the given information in right triangles.