Emilia's car is stuck in the mud. Deandre and Brigeth come along in a truck to help pull her out. They attach one end of a tow strap to the front of the car and the other end to the truck's trailer hitch, and the truck starts to pull. Meanwhile, Deandre and Brigeth get behind the car and push. The truck generates a horizontal force of 318 lb on the car. Deandre and Brigeth are pushing at a slight upward angle and generate a force of 138 lb on the car. These forces can be represented by vectors, as shown in the figure below. The angle between these vectors is 23.6°. Find the resultant force (the vector sum), then give its magnitude and its direction angle from the positive x-axis.

This problem involves finding the resultant force from two vectors that represent the forces applied by the truck and by Deandre and Brigeth. Given that the angle between the two vectors is 23.6°, we can use vector addition techniques to solve for the magnitude and direction of the resultant force.

Here’s how to solve it step-by-step:

### 1. Identify the components:
- **Truck force**: \( F_{\text{truck}} = 318 \, \text{lb} \)
- **Push force**: \( F_{\text{push}} = 138 \, \text{lb} \)
- **Angle between forces**: \( \theta = 23.6^\circ \)

### 2. Apply the Law of Cosines to find the magnitude of the resultant force:

The formula for the magnitude of the resultant force when two vectors are applied at an angle is:

\[
F_{\text{resultant}} = \sqrt{F_{\text{truck}}^2 + F_{\text{push}}^2 + 2 \cdot F_{\text{truck}} \cdot F_{\text{push}} \cdot \cos(\theta)}
\]

Substitute the given values:

\[
F_{\text{resultant}} = \sqrt{318^2 + 138^2 + 2 \cdot 318 \cdot 138 \cdot \cos(23.6^\circ)}
\]

### 3. Find the direction angle using the Law of Sines or by calculating the components:

First, break each force into its horizontal and vertical components, then use trigonometric functions to find the angle of the resultant force with respect to the positive x-axis.

Let’s compute the magnitude and direction of the resultant force.The magnitude of the resultant force is approximately **447.88 lb**, and the direction angle of the resultant force from the positive x-axis is approximately **7.09°**.

Would you like further details or clarifications on this? Here are some related questions to explore:

1. How would the resultant force change if the angle between the vectors increased?
2. What is the significance of the direction angle in relation to the forces applied?
3. How would the forces affect the car’s motion if the truck was pulling at an upward angle instead of horizontally?
4. What happens to the magnitude of the resultant force if the forces applied are perfectly aligned?
5. How does the force magnitude vary if Deandre and Brigeth push with more force?

**Tip:** When dealing with vector addition, breaking forces into components makes it easier to understand their combined effects.

Learn how to calculate the resultant force and direction angle when two forces are applied at an angle. This problem involves vector addition, the Law of Cosines, and trigonometric functions, and is suitable for students in Grades 10-12 studying physics and trigonometry.

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