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To solve for the angle of the reaction force at point \( A \), we need to consider the components of the reaction at \( A \), given the forces acting on the system. We already know the magnitudes of the horizontal and vertical components of the external forces (\( X = 42 \, \text{lb} \) and \( Y = 50 \, \text{lb} \)) and that the reaction at \( A \) consists of both a horizontal and vertical component.

Let's outline the steps to determine the angle:

### Step 1: Define Forces
- \( X = 42 \, \text{lb} \) (horizontal force in the \( -x \)-direction)
- \( Y = 50 \, \text{lb} \) (vertical force in the \( -y \)-direction)
- Reaction at \( A \) can be broken down into:
  - \( A_x \): the horizontal component
  - \( A_y \): the vertical component

### Step 2: Use Static Equilibrium Equations
We already know the reaction magnitudes from the diagram:
- \( A = 46.935 \, \text{lb} \)
- \( B = 31.5 \, \text{lb} \)

For static equilibrium, we have:
1. \( \Sigma F_x = 0 \)
2. \( \Sigma F_y = 0 \)
3. \( \Sigma M_A = 0 \) (taking moments about point \( A \))

Since moments were already used to calculate the reaction forces, we will focus on the horizontal and vertical force equilibrium to determine the angle at \( A \).

From equilibrium:
- \( A_x = X \) (horizontal component of the reaction at \( A \))
- \( A_y = B - Y \) (vertical component of the reaction at \( A \))

### Step 3: Calculate the Angle
The angle \( \theta \) of the reaction at \( A \) is given by the tangent of the ratio between the vertical and horizontal components of the reaction at \( A \):
\[
\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)
\]
Using the magnitudes:
- \( A_x = 42 \, \text{lb} \) (horizontal component)
- \( A_y = 50 \, \text{lb} \) (vertical component)

Let me calculate the angle \( \theta \).The angle of the reaction force at point \( A \) is approximately \( 49.97^\circ \).

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How would the angle change if the force \( X \) increased?
2. What would happen to the reaction forces if the applied force \( Y \) doubled?
3. How do the locations of the applied forces affect the reactions at \( A \) and \( B \)?
4. What is the significance of using moments to find reaction forces?
5. How could the reaction force at \( B \) change if the support type were different?

**Tip:** Understanding how to break forces into components helps simplify many static equilibrium problems!

Learn how to calculate the angle of the reaction force at point A in a statics problem involving forces X = 42 lb and Y = 50 lb. This problem uses key statics principles and trigonometric formulas to solve for the reaction forces at A and B.

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