A cone is hollowed out of a solid wooden cube of side 6 cm. The diameter and height of the cone is same as the side of the cube. What is the volume of the remaining cube ?

Options

A. 159.43 cubic cm
B. 323.15 cubic cm
C. 106.33 cubic cm
D. 210.66 cubic cm

Answer

To find the volume of the remaining cube after hollowing out a cone, we will follow these steps:

### Step 1: Calculate the Volume of the Cube
The volume \( V \) of a cube is given by the formula:
\[ V = a^3 \]
where \( a \) is the length of a side of the cube.

Given that the side of the cube is 6 cm:
\[ V_{\text{cube}} = 6^3 = 216 \text{ cm}^3 \]

### Step 2: Determine the Dimensions of the Cone
The cone that is hollowed out has:
- Height \( h = 6 \) cm (same as the side of the cube)
- Diameter \( d = 6 \) cm, which gives a radius \( r \) of:
\[ r = \frac{d}{2} = \frac{6}{2} = 3 \text{ cm} \]

### Step 3: Calculate the Volume of the Cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values for \( r \) and \( h \):
\[ V_{\text{cone}} = \frac{1}{3} \pi (3^2)(6) = \frac{1}{3} \pi (9)(6) = \frac{54\pi}{3} = 18\pi \text{ cm}^3 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V_{\text{cone}} \approx 18 \times \frac{22}{7} \approx \frac{396}{7} \approx 56.57 \text{ cm}^3 \]

### Step 4: Calculate the Volume of the Remaining Cube
To find the volume of the remaining part of the cube after the cone is hollowed out, we subtract the volume of the cone from the volume of the cube:
\[ V_{\text{remaining}} = V_{\text{cube}} - V_{\text{cone}} \]
Substituting the values:
\[ V_{\text{remaining}} = 216 - 56.57 \approx 159.43 \text{ cm}^3 \]

### Final Answer
The volume of the remaining cube is approximately:
\[ \boxed{159.43 \text{ cm}^3} \]