The number of vertices (V), edges (E) and faces (F) of a polyhedron are respectively 10, 15 and x.
Then, the value of (3x-12) is :

Q: The number of vertices (V), edges (E) and faces (F) of a polyhedron are respectively 10, 15 and x. Then, the value of (3x-12) is :

To determine the value of $3x - 12$ for the given polyhedron, we can use Euler's formula for polyhedra, which states: \[ V - E + F = 2 \] where $ V $ is the number of vertices, $ E $ is the number of edges, and $ F $ is the number of faces. According to the problem, we have: - $ V = 10 $ - $ E = 15 $ - $ F = x $ Substituting these values into Euler's formula, we get: \[ 10 - 15 + x = 2 \] Simplifying this equation: \[ -5 + x = 2 \] Adding 5 to both sides gives: \[ x = 7 \] Now, we need to find the value of $3x - 12$: \[ 3x - 12 = 3(7) - 12 = 21 - 12 = 9 \] Thus, the value of $3x - 12$ is 9, which corresponds to Option 2. ### Explanation of Why Option 2 is Correct: The calculation above shows that when we apply Euler's formula to the given values of vertices and edges, we find that $x = 7$. Substituting $x = 7$ into the expression $3x - 12$ results in 9. Therefore, Option 2 is the correct answer. ### Explanation of Why Other Options are Incorrect: 1. **Option 1 (7):** If $3x - 12$ were equal to 7, then solving $3x - 12 = 7$ would give $x = \frac{19}{3}$, which is not an integer. This contradicts the integer nature of the number of faces in a polyhedron. 2. **Option 3 (14):** If $3x - 12$ were equal to 14, then solving $3x - 12 = 14$ would give $x = \frac{26}{3}$, which is also not an integer. This is not possible for the number of faces. 3. **Option 4 (18):** If $3x - 12$ were equal to 18, then solving $3x - 12 = 18$ would give $x = 10$. Substituting $x = 10$ back into Euler's formula would result in $10 - 15 + 10 = 5$, which does not satisfy Euler's formula $V - E + F = 2$. Therefore, the only option that satisfies all conditions and calculations is Option 2, which is 9.

Question

Accepted Answer

To determine the value of $3x - 12$ for the given polyhedron, we can use Euler's formula for polyhedra, which states:

$$ V - E + F = 2 $$

where $ V $ is the number of vertices, $ E $ is the number of edges, and $ F $ is the number of faces. According to the problem, we have:

- $ V = 10 $
- $ E = 15 $
- $ F = x $

Substituting these values into Euler's formula, we get:

$$ 10 - 15 + x = 2 $$

Simplifying this equation:

$$ -5 + x = 2 $$

Adding 5 to both sides gives:

$$ x = 7 $$

Now, we need to find the value of $3x - 12$:

$$ 3x - 12 = 3(7) - 12 = 21 - 12 = 9 $$

Thus, the value of $3x - 12$ is 9, which corresponds to Option 2.

### Explanation of Why Option 2 is Correct:

The calculation above shows that when we apply Euler's formula to the given values of vertices and edges, we find that $x = 7$. Substituting $x = 7$ into the expression $3x - 12$ results in 9. Therefore, Option 2 is the correct answer.

### Explanation of Why Other Options are Incorrect:

1. **Option 1 (7):** If $3x - 12$ were equal to 7, then solving $3x - 12 = 7$ would give $x = \frac{19}{3}$, which is not an integer. This contradicts the integer nature of the number of faces in a polyhedron.

2. **Option 3 (14):** If $3x - 12$ were equal to 14, then solving $3x - 12 = 14$ would give $x = \frac{26}{3}$, which is also not an integer. This is not possible for the number of faces.

3. **Option 4 (18):** If $3x - 12$ were equal to 18, then solving $3x - 12 = 18$ would give $x = 10$. Substituting $x = 10$ back into Euler's formula would result in $10 - 15 + 10 = 5$, which does not satisfy Euler's formula $V - E + F = 2$.

Therefore, the only option that satisfies all conditions and calculations is Option 2, which is 9.