Answer these two questions based on the following information.
Khayamati is not just a foodie and extravagant but also a party animal. On her birthday her friends got an elephant size cake that was supposed to be disturbed among all her friends. The cake was cylindrical in shape. To cut this cake they got a very large knife which was able to cross through the whole cake from any direction or angle. That is a single cut would be enough to make two pieces of the whole cake. By the way, they made n cuts to the cake from any arbitrary direction they could make. It was notable that until all the n cuts were done no one removed even a single piece.
If n=6, find the maximum possible number of pieces that can be sliced out of that cake.

Options

A. 32
B. 62
C. 42
D. 22

Answer

To solve the problem of determining the maximum possible number of pieces that can be sliced from a cylindrical cake after making \( n = 6 \) cuts, we will use the formula for the cake number in three dimensions.

### Step-by-Step Solution:

1. **Understanding the Problem**: We have a cylindrical cake and we need to find out how many pieces can be created with 6 cuts. Each cut can be made in any direction or angle.

2. **Using the Cake Number Formula**: The maximum number of pieces \( P(n) \) that can be created with \( n \) cuts in a three-dimensional object (like our cylindrical cake) is given by the formula:
\[ P(n) = \frac{n^3 + 5n + 6}{6} \]

3. **Substituting \( n = 6 \)**: We will substitute \( n = 6 \) into the formula:
\[ P(6) = \frac{6^3 + 5 \times 6 + 6}{6} \]

4. **Calculating \( 6^3 \)**:
\[ 6^3 = 216 \]

5. **Calculating \( 5 \times 6 \)**:
\[ 5 \times 6 = 30 \]

6. **Adding the Values**:
\[ 216 + 30 + 6 = 252 \]

7. **Dividing by 6**:
\[ P(6) = \frac{252}{6} = 42 \]

8. **Conclusion**: The maximum possible number of pieces that can be sliced out of the cake with 6 cuts is \( 42 \).

### Final Answer:
The answer is \( 42 \).

---