Using Euler's formula, find the value of P, Q and R respectively.
Faces : P, 5, 20
Vertices : 6, Q, 12
Edges : 12, 9, R

Options

A. 7, 5, 25
B. 8, 6, 30
C. 12, 9, 21
D. 20, 2, 18

Answer

To solve the problem using Euler's formula, we will follow these steps:

### Step 1: Understand Euler's Formula
Euler's formula states that for any convex polyhedron, the relationship between the number of faces (F), vertices (V), and edges (E) is given by:
\[ F + V - E = 2 \]

### Step 2: Set Up the Equations
We have the following information:
- Faces: \( F = P, 5, 20 \)
- Vertices: \( V = 6, Q, 12 \)
- Edges: \( E = 12, 9, R \)

We will create three equations based on Euler's formula for each set of values.

### Step 3: Solve for P
Using the first set of values:
- \( F = P \)
- \( V = 6 \)
- \( E = 12 \)

Substituting into Euler's formula:
\[ P + 6 - 12 = 2 \]

Now, simplify:
\[ P - 6 = 2 \]
\[ P = 2 + 6 \]
\[ P = 8 \]

### Step 4: Solve for Q
Using the second set of values:
- \( F = 5 \)
- \( V = Q \)
- \( E = 9 \)

Substituting into Euler's formula:
\[ 5 + Q - 9 = 2 \]

Now, simplify:
\[ Q - 4 = 2 \]
\[ Q = 2 + 4 \]
\[ Q = 6 \]

### Step 5: Solve for R
Using the third set of values:
- \( F = 20 \)
- \( V = 12 \)
- \( E = R \)

Substituting into Euler's formula:
\[ 20 + 12 - R = 2 \]

Now, simplify:
\[ 32 - R = 2 \]
\[ -R = 2 - 32 \]
\[ -R = -30 \]
\[ R = 30 \]

### Final Values
We have found:
- \( P = 8 \)
- \( Q = 6 \)
- \( R = 30 \)

### Conclusion
Thus, the values of \( P, Q, \) and \( R \) are \( 8, 6, \) and \( 30 \) respectively.

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