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This problem requires us to find the tension \( T \) in the two guy wires, given that the resultant of the three tensions in the wires is vertical. Here's the general approach to solve it:

### Given:
- There are three tensions: two unknown tensions \( T \) and one tension \( B \), which is along the vertical tower.
- The angles are \( 30^\circ \), \( 60^\circ \), and \( 45^\circ \), respectively.
- The wires and the tower are in the same vertical plane, meaning the system is two-dimensional.

### Solution:
We need to apply equilibrium conditions, both in the horizontal and vertical directions, because the system is static.

#### 1. Horizontal Equilibrium:
The sum of the horizontal components of the tensions must be zero since there is no movement in the horizontal direction.
\[
T \cos(30^\circ) + T \cos(60^\circ) = B \cos(45^\circ)
\]
However, because the tension in the vertical wire is vertical, its horizontal component is zero, so we ignore the \( B \) component on the right-hand side:
\[
T \cos(30^\circ) + T \cos(60^\circ) = 0
\]

#### 2. Vertical Equilibrium:
The sum of the vertical components of the tensions must equal the downward tension in the third wire:
\[
T \sin(30^\circ) + T \sin(60^\circ) + B = 0
\]

Next, we can solve for \( T \).

Let me proceed with solving this system.From the horizontal equilibrium equation:

\[
T \cos(30^\circ) + T \cos(60^\circ) = 0
\]

The sum of the horizontal components is approximately \( 1.366 \), which means the horizontal components of the forces balance each other.

For the vertical equilibrium:

\[
T \sin(30^\circ) + T \sin(60^\circ) + B = 0
\]

The sum of the vertical components is also approximately \( 1.366 \).

To proceed, we would need the magnitude of \( B \), the vertical force, to fully solve for \( T \). Could you provide more details about \( B \), or should I assume a standard condition (e.g., the system is symmetric)?

---

Would you like further clarification or details on the calculations?  
Here are 5 follow-up questions to consider:

1. How would the solution change if the angles were different?
2. Can we use this method to solve for non-vertical towers?
3. What would happen if the tensions \( T \) were not equal?
4. How do we interpret the physical meaning of each term in the equilibrium equations?
5. Could we solve this using vector decomposition?

**Tip:** Understanding equilibrium conditions is key when solving for tensions in static systems!

The resultant of three tensions in the guy wires anchored at the top of the tower is vertical. Find the unknown but equal tensions T in the two wires. All three wires and the tower are in the same vertical plane. The angles from left to right are 30 degrees, 60 degrees, and 45 degrees.

Learn how to solve for the tensions in two guy wires anchoring a tower in a static system. This problem involves trigonometry and the equilibrium of forces with angles of 30°, 60°, and 45°. Suitable for engineering students.

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