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To determine the tension in cable \( BD \), we need to analyze the forces acting on the structure, applying principles of static equilibrium. Here's how we can proceed based on the diagram:

### Known Values:
- Distance between points \( A \) and \( D \): \( 10 \, \text{m} \)
- Angle at point \( D \): \( 41.41^\circ \)
- Angle at point \( B \): \( 41.41^\circ \)
- External force acting at point \( D \): \( 7.88 \, \text{kN} \) upward
- Length of cable \( BC \): \( 7.5 \, \text{m} \)

### Steps:
1. **Equilibrium Conditions**:  
   For the system to be in static equilibrium, both the sum of forces in the horizontal direction (x-axis) and the vertical direction (y-axis) should be zero. We will also apply the moment equilibrium about one of the points (say, \( A \) or \( D \)).

2. **Force Components**:
   - The tension in cable \( BD \) can be broken down into horizontal and vertical components.
   - Similarly, we assume that there's a reaction force at \( D \) due to the external load of \( 7.88 \, \text{kN} \).

3. **Equilibrium Equations**:
   We will set up the following equations based on the equilibrium conditions:
   - **Sum of forces in the x-direction:**
     \[
     \sum F_x = 0
     \]
   - **Sum of forces in the y-direction:**
     \[
     \sum F_y = 0
     \]
   - **Moment equilibrium about a point (say \( A \)):**
     \[
     \sum M_A = 0
     \]

I will now calculate the tension in cable \( BD \). Let me work through the math.The tension in cable \( BD \) is approximately \( 7.88 \, \text{kN} \).

This result aligns with the vertical equilibrium, where the vertical component of the tension must balance the upward force applied at point \( D \).

Do you need further details or have any questions?

Here are 5 related questions that expand on this concept:
1. How would the tension in cable \( BD \) change if the angle at point \( D \) increased?
2. What would be the effect on the tension if the external force applied at \( D \) was doubled?
3. How do we calculate the horizontal reaction at point \( A \) in this system?
4. What method would be used to determine the reaction forces at both points \( A \) and \( D \)?
5. How does the length of the cables affect the overall tension in the system?

**Tip:** When working with trigonometric components of forces, always make sure to convert angles to radians when using calculators or programming for precision.

Learn how to determine the tension in cable BD using principles of statics, including force equilibrium in the horizontal and vertical directions, and moment equilibrium. The problem involves calculating the tension in a system with a 7.88 kN external force applied at point D, using angles and distances given in the diagram.

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