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The problem given involves calculating the resultant force from several forces \( F_1, F_2, F_3, F_4 \) with specified magnitudes, angles, and coordinates. Here is a breakdown:

### Given Forces:
1. \( F_1 = 3 \text{ tons}, \alpha_1 = 45^\circ \) at coordinate \( (0,4) \)
2. \( F_2 = 2 \text{ tons}, \alpha_2 = 19^\circ \) at coordinate \( (4,9) \)
3. \( F_3 = 4 \text{ tons}, \alpha_3 = 27^\circ \) at coordinate \( (9,7) \)
4. \( F_4 = 8 \text{ tons}, \alpha_4 = 14^\circ \) at coordinate \( (21,16) \)

### Required:
Find the resultant force using two methods:
a. **Graphically** (using a parallelogram or other vector drawing method)
b. **Mathematically** (by breaking forces into components and summing)

### Steps to Solve:
1. **Resolve each force into its horizontal (\(F_x\)) and vertical (\(F_y\)) components:**
   \[
   F_{x} = F \cdot \cos(\alpha)
   \]
   \[
   F_{y} = F \cdot \sin(\alpha)
   \]
   
   For each force \(F_1, F_2, F_3, F_4\), calculate the components using the given angles and magnitudes.

2. **Sum the components**:
   \[
   R_x = F_{1x} + F_{2x} + F_{3x} + F_{4x}
   \]
   \[
   R_y = F_{1y} + F_{2y} + F_{3y} + F_{4y}
   \]
   
3. **Calculate the resultant force** using the Pythagorean theorem:
   \[
   R = \sqrt{R_x^2 + R_y^2}
   \]

4. **Find the angle** of the resultant force using:
   \[
   \theta = \tan^{-1} \left( \frac{R_y}{R_x} \right)
   \]

Would you like me to proceed with the calculation of components and resultant force? If so, I will go through the detailed steps.

### Related Questions:
1. How do you resolve forces into their components using trigonometry?
2. What is the geometric interpretation of the resultant force using a parallelogram method?
3. How can you find the angle of a resultant force using inverse trigonometry?
4. What role do coordinates play in determining the placement of forces?
5. What are other methods to graphically find the resultant force besides a parallelogram?

### Tip:
When solving for resultant forces, always check the signs of the components (positive or negative) to ensure accuracy in direction.

Find the resultant force from F1, F2, F3, and F4, given their magnitudes, angles, and coordinates.

Learn how to find the resultant force from four different forces given their magnitudes, angles, and coordinates. Solve the problem graphically using the parallelogram method and mathematically by resolving the forces into components. Suitable for college-level physics students.

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