If a solid cone of volume `27pi cm^(3)` is kept inside a hollow cylinder whose radius and height are that of the cone, then the volume of water needed to fill the empty space is

Options

A. `3pi cm^(3)`
B. `18 pi cm^(3)`
C. `54 pi cm^(3)`
D. `81 pi cm^(3)`

Answer

To solve the problem, we need to find the volume of water required to fill the empty space in a hollow cylinder that contains a solid cone. The cone has a volume of \(27\pi \, \text{cm}^3\), and the cylinder has the same radius and height as the cone.

### Step-by-Step Solution:

1. **Understand the Volume of the Cone**:
The volume of the cone is given as \(27\pi \, \text{cm}^3\).

2. **Volume of the Cone Formula**:
The formula for the volume of a cone is given by:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
where \(r\) is the radius and \(h\) is the height of the cone.

3. **Volume of the Cylinder Formula**:
The volume of the cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Since the radius and height of the cylinder are the same as those of the cone, we can express the volume of the cylinder in terms of the cone's volume.

4. **Relate the Volumes**:
Since the volume of the cone is \(27\pi\), we can find the volume of the cylinder:
\[ V_{\text{cylinder}} = 3 \times V_{\text{cone}} = 3 \times 27\pi = 81\pi \, \text{cm}^3 \]

5. **Calculate the Volume of Water**:
The volume of water needed to fill the empty space in the cylinder is the volume of the cylinder minus the volume of the cone:
\[ V_{\text{water}} = V_{\text{cylinder}} - V_{\text{cone}} = 81\pi - 27\pi = 54\pi \, \text{cm}^3 \]

### Final Answer:
The volume of water needed to fill the empty space is \(54\pi \, \text{cm}^3\).

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