Rafaela's car is stuck in the mud. William and Tanya 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, William and Tanya get behind the car and push. The truck generates a horizontal force of 305 lb on the car. William and Tanya are pushing at a slight upward angle and generate a force of 131 lb on the car. These forces can be represented by vectors, as shown in the figure below. The angle between these vectors is 16.9°. Find the resultant force (the vector sum), then give its magnitude and its direction angle from the positive x-axis.


To find the magnitude and direction of the resultant force, we need to use vector addition principles. The two forces can be represented as vectors: the force from the truck and the combined force from William and Tanya.

### Step 1: Components of the forces
- The truck generates a horizontal force of \( F_1 = 305 \, \text{lb} \), which is entirely along the positive \( x \)-axis.
- William and Tanya generate a force of \( F_2 = 131 \, \text{lb} \) at an angle \( \theta = 16.9^\circ \) from the direction of the truck's force.

### Step 2: Resolve forces into components
For \( F_1 \), since it's along the \( x \)-axis:
- \( F_1^x = 305 \, \text{lb} \)
- \( F_1^y = 0 \, \text{lb} \)

For \( F_2 \), we need to resolve it into \( x \) and \( y \) components using trigonometry:
- \( F_2^x = F_2 \cos(\theta) = 131 \cos(16.9^\circ) \)
- \( F_2^y = F_2 \sin(\theta) = 131 \sin(16.9^\circ) \)

### Step 3: Sum the components of the forces
The total force in the \( x \)-direction is:
\[
F_x = F_1^x + F_2^x
\]
The total force in the \( y \)-direction is:
\[
F_y = F_1^y + F_2^y
\]

### Step 4: Magnitude of the resultant force
The magnitude \( F \) of the resultant force is given by the Pythagorean theorem:
\[
F = \sqrt{F_x^2 + F_y^2}
\]

### Step 5: Direction of the resultant force
The direction angle \( \alpha \) (with respect to the positive \( x \)-axis) is:
\[
\alpha = \tan^{-1}\left(\frac{F_y}{F_x}\right)
\]

Let's calculate these values.The magnitude of the resultant force is approximately 432.02 lb, and the direction angle of the resultant force from the positive \( x \)-axis is about \( 5.06^\circ \).

Would you like more details on the calculations or have any questions?

Here are five related questions:
1. What is vector addition, and why is it important in physics?
2. How does the angle between two vectors affect the resultant force?
3. Can the same vector addition method apply to forces in 3D space?
4. How would the problem change if William and Tanya pushed with more force or at a different angle?
5. What is the role of trigonometric functions in resolving forces?

**Tip**: When adding vectors, always ensure you're working with angles relative to the proper axes for accurate component calculations.

Rafaela's car is stuck in the mud. William and Tanya 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, William and Tanya get behind the car and push. The truck generates a horizontal force of 305 lb on the car. William and Tanya are pushing at a slight upward angle and generate a force of 131 lb on the car. These forces can be represented by vectors. The angle between these vectors is 16.9°. Find the resultant force (the vector sum), then give its magnitude and its direction angle from the positive x-axis.

Find the resultant force when a truck generates a 305 lb force pulling a car, while two people push at an upward angle generating a 131 lb force with a 16.9° angle. Learn how vector addition, trigonometry, and the Pythagorean Theorem are used to calculate the magnitude and direction of the resultant force.

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