Let A be a pyramid on a square base and B be a cube. let a,b and c denote the number of edges, number of faces and number of comers, respectively. Then, the result a = b + c is true for

Options

A. a.only A
B. b.only B
C. c.both A and B
D. d.neither A nor B

Answer

To solve the problem, we need to determine the number of edges (a), faces (b), and corners (c) for both a pyramid with a square base and a cube, and check if the relationship \( a = b + c \) holds true for either shape.

### Step-by-Step Solution:

1. **Identify the Pyramid on a Square Base:**
- A pyramid with a square base has:
- **Edges (a):**
- The base has 4 edges.
- There are 4 edges connecting the apex to each vertex of the base.
- Total edges \( a = 4 + 4 = 8 \).
- **Faces (b):**
- The base is 1 face.
- There are 4 triangular faces connecting the apex to each edge of the base.
- Total faces \( b = 1 + 4 = 5 \).
- **Corners (c):**
- The base has 4 corners.
- There is 1 apex.
- Total corners \( c = 4 + 1 = 5 \).

2. **Check the Relationship for the Pyramid:**
- We need to check if \( a = b + c \):
- \( a = 8 \)
- \( b = 5 \)
- \( c = 5 \)
- Calculate \( b + c = 5 + 5 = 10 \).
- Since \( 8 \neq 10 \), the relationship does not hold for the pyramid.

3. **Identify the Cube:**
- A cube has:
- **Edges (a):**
- A cube has 12 edges.
- **Faces (b):**
- A cube has 6 faces.
- **Corners (c):**
- A cube has 8 corners.

4. **Check the Relationship for the Cube:**
- We need to check if \( a = b + c \):
- \( a = 12 \)
- \( b = 6 \)
- \( c = 8 \)
- Calculate \( b + c = 6 + 8 = 14 \).
- Since \( 12 \neq 14 \), the relationship does not hold for the cube.

5. **Conclusion:**
- Since the relationship \( a = b + c \) does not hold for either the pyramid or the cube, the answer is that neither shape satisfies the equation.

### Final Answer:
The result \( a = b + c \) is true for neither the pyramid nor the cube.