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To determine the magnitude of the sum of the forces \( F_A \) and \( F_B \), we can apply vector addition principles. The magnitudes of the two vectors are given as:

- \( |F_A| = 100 \, \text{lb} \)
- \( |F_B| = 60 \, \text{lb} \)

From the diagram, it appears that the angle between the two forces is \( 40^\circ \).

The magnitude of the resultant force \( |F_A + F_B| \) can be found using the Law of Cosines for vector addition:

\[
|F_A + F_B| = \sqrt{|F_A|^2 + |F_B|^2 + 2 |F_A| |F_B| \cos \theta}
\]

Where:
- \( \theta \) is the angle between the two forces, which is \( 40^\circ \),
- \( |F_A| = 100 \, \text{lb} \),
- \( |F_B| = 60 \, \text{lb} \).

Let me calculate the magnitude.The magnitude of the sum of the forces \( |F_A + F_B| \) is approximately \( 150.97 \, \text{lb} \), rounded to three significant figures.

Would you like more details or have any further questions?

Here are some related questions you might consider:
1. How do vector components affect the resultant force?
2. Can the Law of Cosines be applied in three-dimensional space?
3. How would the angle between the vectors change the resultant force?
4. What are the possible real-life applications of vector addition in engineering?
5. How do we decompose forces into their respective components?

**Tip:** Always visualize forces and their components to better understand their resultant when added geometrically.

Determine the magnitude of the sum of the forces F_A + F_B given that |F_A| = 100 lb, |F_B| = 60 lb, and the angle between them is 40 degrees.

Learn how to calculate the resultant force from the vector addition of two forces F_A and F_B using the Law of Cosines. This problem involves trigonometric principles and vector addition, making it suitable for students in Grades 10-12.

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