The following questions are accompanied by three statements A, B and C. You have to determine which statement(s) is/are necessary/sufficient to answer the question.
Find lateral surface area of a cylinder.
A. Volume of a cone with base same as that of the cylinder and height 30 cm is equal to volume of the cylinder,
B. Circumference of base of the cylinder is 132 cm.
C. Volume of the cylinder is 13860 `cm^(3).`

Options

A. Only A and B together
B. Only A and C together
C. All the three together
D. Any two of the three together

Answer

To find the lateral surface area of a cylinder, we need to determine the height and the radius of the cylinder. The formula for the lateral surface area (curved surface area) of a cylinder is given by:

\[ \text{Lateral Surface Area} = 2 \pi R H \]

Where:
- \( R \) is the radius of the base of the cylinder
- \( H \) is the height of the cylinder

Now, let's analyze the statements provided:

### Statement A:
The volume of a cone with the same base as that of the cylinder and height 30 cm is equal to the volume of the cylinder.

The volume of a cone is given by:

\[ V_{cone} = \frac{1}{3} \pi R^2 H_{cone} \]

Since the height of the cone is 30 cm, we can express this as:

\[ V_{cone} = \frac{1}{3} \pi R^2 \times 30 \]

If this volume is equal to the volume of the cylinder, we can set it equal to the volume of the cylinder:

\[ V_{cylinder} = \pi R^2 H_{cylinder} \]

This means we can derive a relationship between the height of the cylinder and the radius, but we still need more information to find both values.

### Statement B:
The circumference of the base of the cylinder is 132 cm.

The circumference \( C \) of a circle is given by:

\[ C = 2 \pi R \]

From this, we can find the radius \( R \):

\[ R = \frac{C}{2 \pi} = \frac{132}{2 \pi} = \frac{66}{\pi} \]

This gives us the radius of the cylinder.

### Statement C:
The volume of the cylinder is 13860 cm³.

The volume of the cylinder is given by:

\[ V_{cylinder} = \pi R^2 H \]

We can use this information along with the radius obtained from Statement B to find the height of the cylinder.

### Conclusion:
To find the lateral surface area of the cylinder, we need both the radius and the height.

- From Statement B, we can find the radius \( R \).
- From either Statement A or Statement C, we can find the height \( H \).

Thus, we can conclude that at least two statements are necessary to find the lateral surface area of the cylinder. Therefore, the answer is that any two of the three statements (A, B, or C) are sufficient to answer the question.

### Final Answer:
**Any two of the three statements (A, B, or C) are sufficient to find the lateral surface area of the cylinder.**

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