To solve this problem, we need to determine the equivalent resultant force and couple moment acting at point $A$, using the given forces $F_1 = 120 \, \text{lb}, F_2 = 210 \, \text{lb}, F_3 = 180 \, \text{lb}$ and the dimensions shown in the diagram.

Steps:

1. Break each force into components:

   • For $F_1$, assuming the angle of the cable is 45° based on the figure, the components in the $x$- and $y$-directions are: $F_{1x} = F_1 \cos(45^\circ), \quad F_{1y} = F_1 \sin(45^\circ)$ Since the angle is measured from the horizontal: $F_{1x} = 120 \times \frac{1}{\sqrt{2}}, \quad F_{1y} = 120 \times \frac{1}{\sqrt{2}}$

   • $F_2$ acts entirely in the $x$-direction, so: $F_{2x} = 210 \, \text{lb}, \quad F_{2y} = 0$

   • $F_3$ acts entirely in the $y$-direction, so: $F_{3x} = 0, \quad F_{3y} = -180 \, \text{lb}$

2. Find the total force components:

   • The total force in the $x$-direction is: $F_{Rx} = F_{1x} + F_{2x} + F_{3x}$
     • The total force in the $y$-direction is: $F_{Ry} = F_{1y} + F_{2y} + F_{3y}$

3. Calculate the moment about point $A$:

   • The moment is given by $M_A = \sum (F \times d)$, where $d$ is the perpendicular distance from point $A$.
   • For each force:
     • $F_1$ creates a moment about $A$ at a distance of 3 ft (horizontal arm).
     • $F_2$ creates a moment at a distance of 4 ft (vertical arm).
     • $F_3$ creates a moment at 6 ft from $A$.

I will compute these steps next to determine the equivalent system.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through the calculations step by step manually.

Would you like to proceed with manual calculations or have any questions about the process?

Here are 5 related questions to deepen your understanding:

1. How do we resolve forces into their components using trigonometry?
2. What is the significance of calculating moments in static equilibrium problems?
3. How does the direction of a force influence its moment about a point?
4. What is the difference between a force and a couple moment in mechanics?
5. Why is it necessary to find the resultant force in systems of multiple forces?

Tip: Always draw a free-body diagram when solving equilibrium problems to visualize forces and their directions.