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To find the resultant force from the given diagram, we need to calculate the sum of all forces in both the \(x\)- and \(y\)-directions. Let’s break down the forces step by step.

### Given Information:
- \(F_1 = 30 \, \text{lbf}\) at an angle of \(60^\circ\) from the negative \(x\)-axis
- \(F_2 = 18 \, \text{lbf}\) along the positive \(x\)-axis
- \(F_3 = 5 \, \text{lbf}\) along the negative \(y\)-axis
- \(F_4 = 10 \, \text{lbf}\) along the negative \(y\)-axis

### Breaking down the forces into components:

#### Force \(F_1\):
- \(F_1\) makes a \(60^\circ\) angle with the negative \(x\)-axis.
- \(F_{1x} = -F_1 \cos(60^\circ) = -30 \cos(60^\circ)\)
- \(F_{1y} = F_1 \sin(60^\circ) = 30 \sin(60^\circ)\)

#### Force \(F_2\):
- \(F_2\) is along the positive \(x\)-axis.
- \(F_{2x} = 18 \, \text{lbf}\)
- \(F_{2y} = 0\)

#### Force \(F_3\):
- \(F_3\) is along the negative \(y\)-axis.
- \(F_{3x} = 0\)
- \(F_{3y} = -5 \, \text{lbf}\)

#### Force \(F_4\):
- \(F_4\) is along the negative \(y\)-axis.
- \(F_{4x} = 0\)
- \(F_{4y} = -10 \, \text{lbf}\)

### Summing up the components:

#### Sum of \(x\)-components:
\[
\Sigma F_x = F_{1x} + F_{2x} + F_{3x} + F_{4x}
\]

#### Sum of \(y\)-components:
\[
\Sigma F_y = F_{1y} + F_{2y} + F_{3y} + F_{4y}
\]

Now, let's calculate these values and find the magnitude of the resultant force \(F_R\) and its direction.### Results:

1. **Sum of \(x\)-components**: \( \Sigma F_x = 3.0000 \, \text{lbf} \)
2. **Sum of \(y\)-components**: \( \Sigma F_y = 10.9808 \, \text{lbf} \)
3. **Resultant force**: \( F_R = 11.3832 \, \text{lbf} \)
4. **Direction of the resultant**: \( \theta_R = 74.7194^\circ \) (from the positive \(x\)-axis)

The resultant force has a magnitude of **11.3832 lbf** and is directed at an angle of **74.72°** from the positive \(x\)-axis.

Would you like more details or have any questions?

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Here are 5 related questions:
1. How would the resultant force change if \(F_2\) increased?
2. What would happen to the resultant direction if \(F_1\) were doubled?
3. How can we decompose forces into components for other angles?
4. How does the system behave if \(F_3\) acts at a different angle?
5. What is the effect of force direction changes on the resultant magnitude?

**Tip:** Always check that the angle reference for forces aligns with the coordinate axes when resolving components.

Calculate the summation of all forces (both for x and y components). Assume F1 to be perpendicular to F2 and parallel to F3.

This problem demonstrates how to calculate the summation of forces with given vector components in both x and y directions. Forces F1, F2, F3, and F4 are involved, with specific angles and directions. Using trigonometry and vector addition, we solve for the resultant force and its direction.

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