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.