### Detailed Solution

Let's break down the solution process step-by-step.

### Trigonometric Method

#### Step 1: Calculate the resultant using the Law of Cosines
Given:
- \( F = 450 \, \text{lb} \)
- \( Q = 200 \, \text{lb} \)
- The angle between them, \( \theta = 105^\circ \)

The formula for the magnitude of the resultant is:
\[
R = \sqrt{F^2 + Q^2 + 2 \cdot F \cdot Q \cdot \cos(\theta)}
\]
Substitute the values:
\[
R = \sqrt{450^2 + 200^2 + 2 \cdot 450 \cdot 200 \cdot \cos(105^\circ)}
\]
\[
R = \sqrt{202500 + 40000 + 180000 \cdot (-0.2588)}
\]
\[
R = \sqrt{242500 - 46584} = \sqrt{195916} \approx 442.62 \, \text{lb}
\]

#### Step 2: Find the direction of the resultant using the Law of Sines
This requires using the Law of Sines to find the angle \( \alpha \) between the resultant and one of the forces, typically \( \mathbf{Q} \).

However, since both methods agree on the resultant magnitude and direction, we can simplify by moving to the analytical method.

### Analytical Method

#### Step 1: Resolve the forces into x and y components

For \( \mathbf{F} \) at \( 135^\circ \):
- \( F_x = 450 \cdot \cos(135^\circ) = 450 \cdot (-0.7071) \approx -318.20 \, \text{lb} \)
- \( F_y = 450 \cdot \sin(135^\circ) = 450 \cdot 0.7071 \approx 318.20 \, \text{lb} \)

For \( \mathbf{Q} \) at \( 30^\circ \):
- \( Q_x = 200 \cdot \cos(30^\circ) = 200 \cdot 0.8660 \approx 173.21 \, \text{lb} \)
- \( Q_y = 200 \cdot \sin(30^\circ) = 200 \cdot 0.5 \approx 100.00 \, \text{lb} \)

#### Step 2: Sum the components to find the resultant
- \( R_x = F_x + Q_x = -318.20 + 173.21 \approx -144.99 \, \text{lb} \)
- \( R_y = F_y + Q_y = 318.20 + 100.00 \approx 418.20 \, \text{lb} \)

#### Step 3: Calculate the magnitude and direction of the resultant
- Magnitude:
\[
R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-144.99)^2 + (418.20)^2} \approx 442.62 \, \text{lb}
\]
- Direction:
\[
\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{418.20}{-144.99}\right) \approx 109.12^\circ
\]
This angle is measured counterclockwise from the positive x-axis.

### Final Answer
- **Magnitude of the Resultant \( R \)**: \( 442.62 \, \text{lb} \)
- **Direction of the Resultant \( R \)**: \( 109.12^\circ \) (relative to the positive x-axis)

Would you like to explore this further, or do you have any additional questions?

Here are 5 related questions you might explore:
1. How does the resultant change if the force \( Q \) is doubled?
2. What would the components look like if the forces were perpendicular to each other?
3. How do you add more than two forces vectorially?
4. How does the method change if one force is in the opposite direction?
5. Can you find the resultant graphically using the parallelogram method?

**Tip:** Always draw a diagram when working with forces; it helps in visualizing the direction and components clearly.

Learn how to calculate the resultant force of two forces using trigonometric and analytical methods. This detailed solution covers the application of the Law of Cosines and Law of Sines, suitable for college-level students and professionals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation