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We are tasked with determining the horizontal component of the reaction force \( A_x \) at pin \( A \) in this system. The system consists of a beam with a load \( P = 418 \, \text{lb} \) applied at the midpoint of a horizontal section. The beam has a 90-degree bend and rests on a surface at point \( B \), which makes a 30-degree angle with the horizontal. Let’s proceed step by step to calculate the required reaction force.

### Given:
- The total load \( P = 418 \, \text{lb} \) acts vertically at the midpoint of the horizontal section.
- The horizontal distances on either side of the load are 3 ft.
- The vertical height from \( B \) is 2 ft.
- The incline at point \( B \) is 30 degrees.

### Steps:
1. **Establish equilibrium equations** for the beam since it is in static equilibrium. We will use:
   - \( \Sigma F_x = 0 \) for horizontal forces.
   - \( \Sigma F_y = 0 \) for vertical forces.
   - \( \Sigma M_A = 0 \) for the moment about point \( A \).

2. **Determine reactions at \( B \)**: 
   - Let the reaction at \( B \) be \( B_x \) and \( B_y \) (the components of the reaction force at \( B \)).

3. **Moment about point \( A \)**: 
   The moment about \( A \) involves contributions from the force \( P \) and the vertical and horizontal components of the reaction at \( B \). 
   
Let me calculate the reactions based on the given geometry.From the calculations, the reaction force at point \( B \) has the following components:
- \( B_x \), the horizontal component: \( 313.5 \, \text{lb} \)
- \( B_y \), the vertical component: \( 543.0 \, \text{lb} \)

Additionally, the moment due to the load \( P \) about point \( A \) is \( 1254 \, \text{lb-ft} \).

### Next Steps:
1. Apply the sum of horizontal forces \( \Sigma F_x = 0 \) to find \( A_x \) (the horizontal component of the reaction at \( A \)).
   - \( A_x + B_x = 0 \)
   - Therefore, \( A_x = -B_x = -313.5 \, \text{lb} \).

Thus, the horizontal reaction component at point \( A \) is \( \boxed{314 \, \text{lb}} \) (rounded to the nearest whole pound).

Would you like any further details or clarifications?

Here are 5 related questions you might explore:
1. How would the reaction forces change if the load were not placed at the midpoint?
2. What would happen to the reactions if the angle at point \( B \) were different, such as 45 degrees?
3. How would increasing the total load \( P \) affect the horizontal component at \( A \)?
4. Can we calculate the vertical reaction at \( A \) as well?
5. How would the reaction forces change if point \( B \) were fixed instead of being on a roller?

**Tip**: When solving static equilibrium problems, always begin by identifying all forces and moments acting on the system, and use the relevant equilibrium conditions to solve step by step.

Determine the horizontal component of reaction, Ax, on the member at the pin A, when an external load of 418 lb is acting at the midpoint of the beam as shown in the image. Give your answer in lb to the nearest whole lb.

This problem involves calculating the horizontal reaction force (Ax) at pin A in a system where a 418 lb load is applied at the midpoint of the beam. The reaction forces at both A and B are considered, along with equilibrium conditions for forces and moments. Suitable for university-level engineering students or advanced high school physics.

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