https://file.aiquickdraw.com/mathbot/file/1726464816946sumtjytn.jpg

To determine the resultant of the four forces shown in the diagram, we need to break each force into its components along the X and Y axes and then sum the components. Here are the steps:

### Step 1: Identify the Forces and Angles
The forces and their directions are:

1. **Force 1 (448 lb)** directed along the positive Y-axis.
2. **Force 2 (600 lb)** at an angle from the positive X-axis (looks like 30°).
3. **Force 3 (361 lb)** at an angle from the negative X-axis (135°).
4. **Force 4 (260 lb)** at an angle (around 210° from the positive X-axis).

We'll need to resolve each force into its horizontal (X) and vertical (Y) components using trigonometry.

### Step 2: Resolve the Forces into Components
Using the formulas:

\[
F_{x} = F \cdot \cos(\theta)
\]
\[
F_{y} = F \cdot \sin(\theta)
\]

#### Force 1 (448 lb) - Vertical (Y-axis)
Since this force is aligned directly with the Y-axis, it has no X-component.

- \( F_{1x} = 0 \)
- \( F_{1y} = 448 \, \text{lb} \)

#### Force 2 (600 lb) - At 30° from the X-axis
- \( F_{2x} = 600 \cdot \cos(30^\circ) \)
- \( F_{2y} = 600 \cdot \sin(30^\circ) \)

#### Force 3 (361 lb) - At 135° from the X-axis
- \( F_{3x} = 361 \cdot \cos(135^\circ) \)
- \( F_{3y} = 361 \cdot \sin(135^\circ) \)

#### Force 4 (260 lb) - At 210° from the X-axis
- \( F_{4x} = 260 \cdot \cos(210^\circ) \)
- \( F_{4y} = 260 \cdot \sin(210^\circ) \)

### Step 3: Sum the Components
Once the components of all forces are calculated, sum the X-components and Y-components to find the total force components.

\[
R_x = F_{1x} + F_{2x} + F_{3x} + F_{4x}
\]
\[
R_y = F_{1y} + F_{2y} + F_{3y} + F_{4y}
\]

### Step 4: Calculate the Resultant Force
The resultant force \( R \) can be found using:

\[
R = \sqrt{R_x^2 + R_y^2}
\]

The angle of the resultant with respect to the X-axis is:

\[
\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)
\]

Let me calculate these components for you!The resultant force and its components are as follows:

- \( R_x = 39.18 \, \text{lb} \) (X-component)
- \( R_y = 873.27 \, \text{lb} \) (Y-component)
- Resultant force \( R = 874.14 \, \text{lb} \)
- The angle of the resultant with respect to the X-axis is approximately \( 87.43^\circ \)

Would you like a detailed breakdown of how each component was calculated, or do you have any questions about this process?

Here are some related questions to expand your understanding:
1. How do trigonometric functions help in resolving forces?
2. What is the significance of the angle of the resultant force?
3. How does changing one force affect the overall resultant?
4. Can the resultant force be zero, and under what conditions?
5. What would happen if more than four forces were acting on the body?

**Tip:** Always ensure you are using the correct angle in trigonometric functions, especially when forces are at angles not directly along the axes.

Draw and determine the resultant of the four forces acting on the body shown in the figure.

Learn how to calculate the resultant of four forces by resolving them into X and Y components and using vector addition. This example includes forces of 448 lb, 600 lb, 361 lb, and 260 lb acting at various angles.

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