The question involves calculating the distance between point B and point P in a cube $ABCD.EFGH$ where the edge length of the cube is 4 cm.

Step-by-Step Solution:

1. Cube Information: The cube has side lengths of 4 cm. Let's consider point B as the bottom front-right corner and point P as the center of the top face.

2. Coordinates in 3D Space: Let's place the cube in a 3D coordinate system with:

   • $A(0, 0, 0)$,
   • $B(4, 0, 0)$,
   • $D(0, 4, 0)$,
   • $H(0, 4, 4)$, and
   • $P(2, 2, 4)$ (since it's the center of the top face).

3. Distance Formula: To find the distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$, we use the 3D distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

4. Substitute Coordinates of B and P: Point B is $(4, 0, 0)$, and point P is $(2, 2, 4)$. $d = \sqrt{(2 - 4)^2 + (2 - 0)^2 + (4 - 0)^2}$ $d = \sqrt{(-2)^2 + (2)^2 + (4)^2}$ $d = \sqrt{4 + 4 + 16} = \sqrt{24} = 2\sqrt{6}$

Thus, the distance between point B and point P is $2\sqrt{6}$ cm.

Would you like further clarification or additional details?

Here are 5 related questions you can explore:

1. How do you find the diagonal length of a cube?
2. What is the relationship between the face center and cube vertices in 3D geometry?
3. How would the distance change if the cube’s side length was doubled?
4. How do you calculate the distance between any two points in a 3D plane?
5. How does symmetry simplify calculations in geometric solids?

Tip: For complex 3D problems, always sketch the situation and assign coordinates to help visualize the relationships between points!