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To determine the tensions in the two cables holding the 200 N weight, let's call the tensions \( T_1 \) and \( T_2 \) for the left and right cables, respectively. Here’s how we can approach the problem:

1. **Identify forces and angles:**
   - The weight \( W = 200 \, \text{N} \) acts downward.
   - The angles with the horizontal are 40° for \( T_1 \) and 30° for \( T_2 \).

2. **Resolve forces into components:**
   - The sum of the vertical components of \( T_1 \) and \( T_2 \) should equal the weight (since the weight is the only downward force).
   - The horizontal components of \( T_1 \) and \( T_2 \) should cancel each other out (since there is no horizontal movement).

3. **Set up equations:**
   - Vertical components: \( T_1 \sin(40^\circ) + T_2 \sin(30^\circ) = 200 \).
   - Horizontal components: \( T_1 \cos(40^\circ) = T_2 \cos(30^\circ) \).

4. **Solve the equations:**
   From these two equations, we can solve for \( T_1 \) and \( T_2 \).

Let's calculate this step-by-step.

### Calculations
1. **Horizontal Equation:**
   \[
   T_1 \cos(40^\circ) = T_2 \cos(30^\circ)
   \]
   Solving for \( T_1 \):
   \[
   T_1 = T_2 \frac{\cos(30^\circ)}{\cos(40^\circ)}
   \]

2. **Vertical Equation:**
   Substitute \( T_1 \) from the horizontal equation into the vertical equation:
   \[
   T_2 \frac{\cos(30^\circ)}{\cos(40^\circ)} \sin(40^\circ) + T_2 \sin(30^\circ) = 200
   \]
   Solving this will give us \( T_2 \), which can then be used to find \( T_1 \).

I’ll proceed with these calculations to find the values of \( T_1 \) and \( T_2 \).The tensions in the two cables are as follows:

- \( T_1 \approx 184.32 \, \text{N} \) (left cable with 40° angle)
- \( T_2 \approx 163.04 \, \text{N} \) (right cable with 30° angle)

These values satisfy both the vertical and horizontal equilibrium conditions for the system.

Would you like a detailed breakdown of the calculations? Let me know if you have further questions.

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Here are some related questions to deepen your understanding:

1. How would the tensions change if the weight were increased to 300 N?
2. What would happen to the tensions if both angles were equal, say 35° each?
3. How does changing the angle of one of the cables affect the distribution of tension?
4. Could you solve for tensions using a different approach, such as vector analysis?
5. How would adding a third cable affect the equilibrium conditions?

**Tip:** Always check equilibrium conditions (both horizontal and vertical) when solving tension problems to ensure accuracy.

Determinar las tensiones en los cables que sostienen un peso de 200 N con ángulos de 40° y 30° respecto a la horizontal.

Learn how to calculate the tensions in two cables supporting a 200 N weight, with angles of 40° and 30° relative to the horizontal. This problem demonstrates the application of trigonometry and static equilibrium conditions in physics, suitable for high school students in Grades 10-12.

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