A triangular prism has `underline(P)` faces , `underline(Q)` vertices and `underline( R)` edges

Options

A. `{:(P , Q , R) , (5 , 5 , 8):}`
B. `{:(P , Q , R ) , (6 , 8 , 12):}`
C. `{:(P , Q , R ) , (5 , 6 , 8):}`
D. `{:(P , Q , R) , (5 , 6 , 9):}`

Answer

To find the values of \( P \), \( Q \), and \( R \) for a triangular prism, we will determine the number of faces, vertices, and edges step by step.

### Step 1: Count the Faces (P)
A triangular prism consists of:
- 2 triangular faces (top and bottom)
- 3 rectangular faces (the sides connecting the triangles)

So, the total number of faces \( P \) is:
\[ P = 2 \text{ (triangular faces)} + 3 \text{ (rectangular faces)} = 5 \]

### Step 2: Count the Vertices (Q)
A triangular prism has:
- 3 vertices on the top triangular face
- 3 vertices on the bottom triangular face

Thus, the total number of vertices \( Q \) is:
\[ Q = 3 \text{ (top vertices)} + 3 \text{ (bottom vertices)} = 6 \]

### Step 3: Count the Edges (R)
The edges of a triangular prism consist of:
- 3 edges on the top triangular face
- 3 edges on the bottom triangular face
- 3 edges connecting the corresponding vertices of the top and bottom triangles

Therefore, the total number of edges \( R \) is:
\[ R = 3 \text{ (top edges)} + 3 \text{ (bottom edges)} + 3 \text{ (connecting edges)} = 9 \]

### Final Values
Now we have:
- \( P = 5 \)
- \( Q = 6 \)
- \( R = 9 \)

### Step 4: Calculate \( PQR \)
To find the product \( PQR \):
\[ PQR = P \times Q \times R = 5 \times 6 \times 9 \]

Calculating this:
\[ 5 \times 6 = 30 \]
\[ 30 \times 9 = 270 \]

Thus, the final answer for \( PQR \) is:
\[ PQR = 270 \]

### Summary of Values
- \( P = 5 \)
- \( Q = 6 \)
- \( R = 9 \)
- \( PQR = 270 \)