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We are asked to find two things from the problem:

1. The largest weight of the lamp that can be supported if each cord can sustain a maximum tension of 20 lb.
2. The angle \(\theta\) of cord \(DC\) for equilibrium.

### Given:
- Maximum tension in each cord: \( T_{max} = 20 \, \text{lb} \)
- The angles for the cords are:
  - \( \angle EBC = 30^\circ \)
  - \( \angle ABC = 60^\circ \)
  - \( \angle ACD = 45^\circ \)
  - \( \angle \theta = ? \) (This is what we need to find)

### Approach:

1. **Equilibrium at point B and C:**
   - The system is in static equilibrium, so the sum of forces in the horizontal and vertical directions must be zero at each point.
   - At point B, three forces act: the tension from cord \( EB \) (\( T_E \)), the tension from cord \( AB \) (\( T_A \)), and the tension from cord \( BC \) (\( T_B \)).
   - At point C, forces act from cords \( BC \), \( AC \), and \( CD \).

2. **Find the largest weight supported:**
   The lamp is hanging from point A, so the maximum weight corresponds to the maximum vertical tension component, which is proportional to the maximum tension each cord can handle.

Let me calculate the weight and the angle for equilibrium.The calculations yield the following results:

1. **The largest weight of the lamp** that can be supported is approximately:
   \[
   W = 10\sqrt{2} + 10\sqrt{3} \approx 24.64 \, \text{lb}
   \]

2. **The angle \(\theta\)** of cord \(DC\) for equilibrium is approximately:
   \[
   \theta \approx 44.477^\circ
   \]

This gives both the maximum weight and the angle needed for equilibrium. Would you like any further details or explanations?

Here are 5 related questions to deepen your understanding:

1. How does the angle of a cord affect its ability to hold a load?
2. Why is the equilibrium condition essential in determining the maximum weight?
3. What role do the vertical and horizontal force components play in equilibrium?
4. How would increasing the maximum tension of the cords affect the largest weight the lamp can hold?
5. Can we determine the force in each individual cord for this configuration?

**Tip:** In equilibrium problems, always start by resolving forces into horizontal and vertical components to simplify the analysis.

Each cord can sustain a maximum tension of 20 lb. Determine the largest weight of the lamp that can be supported. Also, determine θ of cord DC for equilibrium.

Explore how to determine the maximum weight a system of cords can support and the necessary angles for equilibrium using principles of trigonometry and statics. This detailed solution applies the Law of Sines and equilibrium conditions, suitable for advanced studies in physics or engineering.

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