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.
After cutting the cake if they got total 300 pieces, find the minimum possible number of cuts required.

Options

A. 13
B. 17
C. 11
D. 12

Answer

To solve the problem of finding the minimum number of cuts required to obtain 300 pieces from a cylindrical cake, we can use the formula for the maximum number of regions (pieces) created by n cuts in three-dimensional space. The formula is given by:

\[ R(n) = \frac{n^3 + 5n + 6}{6} \]

Where \( R(n) \) is the maximum number of pieces formed by \( n \) cuts.

### Step-by-step Solution:

1. **Set up the inequality**:
We know that after making \( n \) cuts, the number of pieces \( R(n) \) must be at least 300. Therefore, we can set up the inequality:
\[ R(n) \geq 300 \]
This translates to:
\[ \frac{n^3 + 5n + 6}{6} \geq 300 \]

2. **Multiply both sides by 6**:
To eliminate the fraction, we multiply both sides of the inequality by 6:
\[ n^3 + 5n + 6 \geq 1800 \]

3. **Rearrange the inequality**:
Rearranging gives us:
\[ n^3 + 5n - 1794 \geq 0 \]

4. **Estimate values for n**:
We need to find the smallest integer \( n \) such that the inequality holds. We can test successive integer values for \( n \):

- For \( n = 11 \):
\[ 11^3 + 5 \cdot 11 = 1331 + 55 = 1386 \quad (\text{not sufficient, } 1386 < 1794) \]

- For \( n = 12 \):
\[ 12^3 + 5 \cdot 12 = 1728 + 60 = 1788 \quad (\text{not sufficient, } 1788 < 1794) \]

- For \( n = 13 \):
\[ 13^3 + 5 \cdot 13 = 2197 + 65 = 2262 \quad (\text{sufficient, } 2262 > 1794) \]

5. **Conclusion**:
Since \( n = 13 \) is the first value that satisfies the inequality, the minimum number of cuts required to achieve at least 300 pieces is:
\[ \boxed{13} \]