From a solid wooden cylinder of height 28 cm and diameter 6 cm, two conical cavities are hollowed out. The diameters of the cones are also of 6 cm and height 10.5 cm. Taking `pi=(22)/(7)` find the volume of the remaining solid.

Answer

To find the volume of the remaining solid after hollowing out two conical cavities from a solid wooden cylinder, we will follow these steps:

### Step 1: Calculate the volume of the cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

**Given:**
- Height of the cylinder \( h = 28 \) cm
- Diameter of the cylinder \( d = 6 \) cm, thus the radius \( r = \frac{d}{2} = \frac{6}{2} = 3 \) cm

**Substituting the values:**
\[ V_{\text{cylinder}} = \frac{22}{7} \times (3)^2 \times 28 \]
\[ = \frac{22}{7} \times 9 \times 28 \]
\[ = \frac{22 \times 9 \times 28}{7} \]

### Step 2: Simplify the volume of the cylinder
Calculating \( 9 \times 28 \):
\[ 9 \times 28 = 252 \]
Now substituting back:
\[ V_{\text{cylinder}} = \frac{22 \times 252}{7} \]
Calculating \( \frac{252}{7} \):
\[ 252 \div 7 = 36 \]
Thus:
\[ V_{\text{cylinder}} = 22 \times 36 = 792 \text{ cm}^3 \]

### Step 3: Calculate the volume of one cone
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]

**Given:**
- Height of the cone \( h = 10.5 \) cm
- Diameter of the cone \( d = 6 \) cm, thus the radius \( r = 3 \) cm

**Substituting the values:**
\[ V_{\text{cone}} = \frac{1}{3} \times \frac{22}{7} \times (3)^2 \times 10.5 \]
\[ = \frac{1}{3} \times \frac{22}{7} \times 9 \times 10.5 \]
Calculating \( 9 \times 10.5 \):
\[ 9 \times 10.5 = 94.5 \]
Thus:
\[ V_{\text{cone}} = \frac{1}{3} \times \frac{22 \times 94.5}{7} \]

### Step 4: Simplify the volume of one cone
Calculating \( \frac{22 \times 94.5}{7} \):
\[ 22 \times 94.5 = 2079 \]
Now substituting back:
\[ V_{\text{cone}} = \frac{2079}{3 \times 7} = \frac{2079}{21} \]
Calculating \( 2079 \div 21 \):
\[ 2079 \div 21 = 99 \]
Thus:
\[ V_{\text{cone}} = 99 \text{ cm}^3 \]

### Step 5: Calculate the volume of two cones
Since there are two conical cavities:
\[ V_{\text{2 cones}} = 2 \times V_{\text{cone}} = 2 \times 99 = 198 \text{ cm}^3 \]

### Step 6: Calculate the volume of the remaining solid
Now, we can find the volume of the remaining solid:
\[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{2 cones}} \]
\[ = 792 - 198 = 594 \text{ cm}^3 \]

### Final Answer
The volume of the remaining solid is:
\[ \boxed{594 \text{ cm}^3} \]