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The problem involves calculating the angle between the line \( PB \) and the plane \( ABCD \), which includes a 3D diagram of a wedge structure. Let's break down the information:

### Information from the diagram:
- \( AB = 14 \, \text{cm} \), \( BC = 18 \, \text{cm} \), and the rectangle \( ABCD \) lies in a horizontal plane.
- \( ADEF \) is a rectangle in a vertical plane, with \( AD = 14 \, \text{cm} \).
- \( P \) is one-third of the way from \( E \) to \( F \).
- \( PQ \) is perpendicular to \( AD \).
- The angle between \( PC \) and the plane \( ABCD \) is \( 30^\circ \).

We need to calculate the angle between the line \( PB \) and the plane \( ABCD \). To solve this:

1. **Vectors Setup:**
   - Represent the points in terms of vectors in 3D space with coordinates based on the given lengths.
   
2. **Line \( PB \) and Plane \( ABCD \):**
   - To calculate the angle between a line and a plane, we use the formula:
     \[
     \sin \theta = \frac{\lvert \mathbf{n} \cdot \mathbf{v} \rvert}{\lvert \mathbf{n} \rvert \lvert \mathbf{v} \rvert}
     \]
     where \( \mathbf{n} \) is the normal vector to the plane, and \( \mathbf{v} \) is the direction vector of line \( PB \).

3. **Steps:**
   - Compute the coordinates of points \( P \), \( B \), and vectors.
   - Derive the normal vector to the plane \( ABCD \).
   - Use the dot product to find the angle between the vectors.

Let me calculate this for you.The angle between the line \( PB \) and the plane \( ABCD \) is approximately \( 28.47^\circ \), correct to 3 significant figures.

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you find the normal vector of a plane given three points?
2. What is the dot product, and how is it used to find angles between vectors?
3. How does the placement of point \( P \) (as one-third of the way from \( E \) to \( F \)) affect the calculations?
4. How would the result change if the angle between \( PC \) and the plane \( ABCD \) were different from \( 30^\circ \)?
5. What is the geometric significance of the angle between a line and a plane?

**Tip:** The angle between a line and a plane is calculated using the normal vector to the plane because it simplifies the 3D geometry problem into manageable vector operations.

The diagram shows a wedge. AB = 14 cm and BC = 18 cm. ABCD is a rectangle in a horizontal plane. ADEF is a rectangle in a vertical plane. P is one-third of the way from E to F. PQ is perpendicular to AD. The angle between PC and the plane ABCD is 30°. Calculate the angle between PB and the plane ABCD. Give your answer correct to 3 s.f.

Learn how to calculate the angle between line PB and the plane ABCD using vector geometry. Given a 3D wedge diagram with specific angles and dimensions, this problem incorporates the dot product theorem and trigonometry for a complete solution.

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