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Let's solve the problem step by step.

### Given forces:
- \( F_1 = 164 \, \text{N} \) (to the right, along the x-axis)
- \( F_2 = 286 \, \text{N} \) (to the right, along the x-axis)
- \( F_3 = 263 \, \text{N} \) at \( 60^\circ \) above the horizontal (this force has both x and y components)
- \( F_4 = 161 \, \text{N} \) (to the left, along the negative x-axis)

### Step 1: Resolve each force into components.
- \( F_1 \) is entirely in the positive x-direction, so:  
  \( F_{1x} = 164 \, \text{N}, \quad F_{1y} = 0 \)
  
- \( F_2 \) is entirely in the positive x-direction, so:  
  \( F_{2x} = 286 \, \text{N}, \quad F_{2y} = 0 \)
  
- \( F_3 \) has both x and y components:  
  \( F_{3x} = 263 \times \cos(60^\circ) = 263 \times 0.5 = 131.5 \, \text{N} \)  
  \( F_{3y} = 263 \times \sin(60^\circ) = 263 \times 0.866 = 227.1 \, \text{N} \)
  
- \( F_4 \) is entirely in the negative x-direction, so:  
  \( F_{4x} = -161 \, \text{N}, \quad F_{4y} = 0 \)

### Step 2: Sum the components.
#### X-components:
\[
F_{x,\text{total}} = F_{1x} + F_{2x} + F_{3x} + F_{4x} = 164 + 286 + 131.5 - 161 = 420.5 \, \text{N}
\]
#### Y-components:
\[
F_{y,\text{total}} = F_{1y} + F_{2y} + F_{3y} + F_{4y} = 0 + 0 + 227.1 + 0 = 227.1 \, \text{N}
\]

### Step 3: Find the magnitude of the resultant force.
Using the Pythagorean theorem:
\[
F_{\text{resultant}} = \sqrt{(F_{x,\text{total}})^2 + (F_{y,\text{total}})^2} = \sqrt{(420.5)^2 + (227.1)^2}
\]
\[
F_{\text{resultant}} = \sqrt{176820.25 + 51516.41} = \sqrt{228336.66} = 477.81 \, \text{N}
\]

### Step 4: Find the angle of the resultant force.
The angle \( \theta \) is given by:
\[
\theta = \tan^{-1} \left( \frac{F_{y,\text{total}}}{F_{x,\text{total}}} \right) = \tan^{-1} \left( \frac{227.1}{420.5} \right)
\]
\[
\theta = \tan^{-1}(0.54) = 28.2^\circ
\]

### Final Answer:
The resultant angle of the forces is \( 28.2^\circ \) above the horizontal.

Would you like further details or have any questions?

### Relative Questions:
1. How do you resolve forces into components in general?
2. What is the significance of using trigonometric functions in force resolution?
3. How does changing the angle of \( F_3 \) affect the resultant force?
4. Can the resultant force ever have a negative angle? Why or why not?
5. How would the result differ if one of the forces was removed?

### Tip:
Always break forces into components along standard axes to simplify vector addition.

Four forces are acting on an object. Find the resultant angle of the forces if F1 = 164 N, F2 = 286 N, F3 = 263 N at 60 degrees above the horizontal, and F4 = 161 N.

Learn how to calculate the resultant force and angle for an object subjected to multiple forces, including resolving forces into components and using trigonometry. This problem involves vector addition and is suitable for high school or college students.

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