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Answer this also do the checking the answer must be zero

Let's go through the problem step by step.

### Problem Recap:
- A traffic light is suspended by a cable, with each side of the cable making a \(10^\circ\) angle with the horizontal.
- The maximum safe tension in each cable is \(1.0 \times 10^3 \, \text{N}\).
- We need to determine the maximum safe mass of the traffic light.

### Step-by-Step Solution:

1. **Free Body Diagram and Forces:**
   - There are two cables holding up the traffic light, both making an angle of \(10^\circ\) with the horizontal.
   - The weight \(W\) of the traffic light acts vertically downward, and each cable has a tension \(T\) acting along the direction of the cable.
   
2. **Vertical Force Balance:**
   Since the traffic light is in equilibrium, the sum of the vertical components of the tension forces must equal the weight:
   \[
   2T \sin(10^\circ) = W
   \]
   where \(W = mg\), \(m\) is the mass of the traffic light, and \(g \approx 9.8 \, \text{m/s}^2\).

3. **Horizontal Force Balance:**
   The horizontal components of the tensions must cancel each other out:
   \[
   T \cos(10^\circ) = T \cos(10^\circ)
   \]
   This equation is always true and doesn't give us new information for solving the problem.

4. **Solving for the Mass:**
   From the vertical balance equation:
   \[
   W = 2T \sin(10^\circ)
   \]
   Substituting \(W = mg\), we get:
   \[
   mg = 2T \sin(10^\circ)
   \]
   Solving for \(m\):
   \[
   m = \frac{2T \sin(10^\circ)}{g}
   \]
   Substituting the values \(T = 1.0 \times 10^3 \, \text{N}\), \(\sin(10^\circ) \approx 0.1736\), and \(g = 9.8 \, \text{m/s}^2\):
   \[
   m = \frac{2(1.0 \times 10^3)(0.1736)}{9.8}
   \]
   \[
   m = \frac{347.2}{9.8} \approx 35.43 \, \text{kg}
   \]

### Checking the Answer:
Let's check if the units and values make sense:
- The calculated mass is \(m \approx 35.43 \, \text{kg}\).
- The tension limit of \(1.0 \times 10^3 \, \text{N}\) is enough to support this mass.

The tension forces balance properly, and no forces are left unaccounted for, meaning the answer checks out.

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Would you like any further details or clarifications? Here are five related questions that could expand on this topic:

1. What happens to the maximum safe mass if the angle increases?
2. How would the tension change if the mass of the light increased by 50%?
3. What is the horizontal force exerted by each cable?
4. How would the system behave if only one cable supported the traffic light?
5. Could wind or other external forces impact the tension in the cables?

**Tip:** When analyzing forces in equilibrium, always break the forces into vertical and horizontal components to ensure all forces are balanced properly.

A traffic light is supported by a cable that makes an angle of 10° with the horizontal on each side of the light. The maximum safe tension in the cable is 1.0 × 10³ N. What is the maximum safe mass for the light?

Calculate the maximum safe mass of a traffic light supported by cables at a 10° angle with a maximum tension of 1.0 × 10³ N. This problem uses trigonometry and Newton's laws of motion, suitable for Grades 10-12.

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