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To solve the problem, we need to find the force \( w \) required to make the component of the force parallel to the inclined plane equal to 2.2 lb.

### Key Information:
- The angle between the force \( w \) and the inclined plane is \( 32^\circ \).
- The incline angle of the plane is \( 15^\circ \).
- The parallel force component must be \( 2.2 \, \text{lb} \).

### Force Decomposition:
The force \( w \) can be decomposed into two components:
1. One component parallel to the inclined plane.
2. One component perpendicular to the inclined plane.

The parallel component is given by the formula:
\[
F_{\parallel} = w \cos(\theta)
\]
Where:
- \( \theta = 32^\circ - 15^\circ = 17^\circ \) (this is the angle between \( w \) and the direction parallel to the incline).
- \( F_{\parallel} = 2.2 \, \text{lb} \) (as given).

### Step-by-Step Calculation:
\[
F_{\parallel} = w \cos(17^\circ)
\]
\[
2.2 = w \cos(17^\circ)
\]
\[
w = \frac{2.2}{\cos(17^\circ)}
\]

Let me calculate that for you.The force \( w \) needed is approximately \( 2.30 \, \text{lb} \), rounded to the nearest hundredth.

Would you like more details on this calculation or have any questions?

Here are some related questions:
1. How do we calculate the perpendicular component of the force in this situation?
2. What happens if the incline angle increases while keeping the parallel component constant?
3. How do we determine the total force acting on the box if friction is involved?
4. Can we apply the same method if the force \( w \) is applied at a different angle?
5. How does the weight of the object affect the force required to move it up the incline?

**Tip:** Remember that force components can be decomposed into perpendicular and parallel directions to better analyze motion on inclined planes.

Suppose that a box is being towed up an inclined plane as shown in the figure. Find the force w needed to make the component of the force parallel to the inclined plane equal to 2.2 lb.

Learn how to calculate the force needed to make the component of force parallel to the inclined plane equal to 2.2 lb. Using trigonometric force decomposition with angles of 15° and 32°, this problem involves understanding the relationship between forces and angles.

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