From a rectangular solid of metal 42 cm by 30 cm by 20 cm, a conical cavity of diameter 14 cm and depth 24 cm is drilled out. Find :
the volume of remaining solid,

Answer

To find the volume of the remaining solid after drilling out a conical cavity from a rectangular solid, we will follow these steps:

### Step 1: Calculate the Volume of the Rectangular Solid
The volume \( V \) of a rectangular solid can be calculated using the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
Given:
- Length = 42 cm
- Breadth = 20 cm
- Height = 30 cm

Substituting the values:
\[ V = 42 \, \text{cm} \times 20 \, \text{cm} \times 30 \, \text{cm} \]
\[ V = 25200 \, \text{cm}^3 \]

### Step 2: Calculate the Volume of the Conical Cavity
The volume \( V \) of a cone can be calculated using the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( r \) is the radius of the base of the cone
- \( h \) is the height of the cone

Given:
- Diameter of the cone = 14 cm, so the radius \( r = \frac{14}{2} = 7 \, \text{cm} \)
- Height \( h = 24 \, \text{cm} \)

Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{1}{3} \times \frac{22}{7} \times (7 \, \text{cm})^2 \times 24 \, \text{cm} \]
Calculating \( (7 \, \text{cm})^2 \):
\[ (7 \, \text{cm})^2 = 49 \, \text{cm}^2 \]
Now substituting this back into the volume formula:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 49 \, \text{cm}^2 \times 24 \, \text{cm} \]
\[ V = \frac{1}{3} \times 22 \times 7 \times 24 \, \text{cm}^3 \]
Calculating:
\[ V = \frac{1}{3} \times 22 \times 168 \, \text{cm}^3 \]
\[ V = \frac{3696}{3} \, \text{cm}^3 = 1232 \, \text{cm}^3 \]

### Step 3: Calculate the Volume of the Remaining Solid
To find the volume of the remaining solid, we subtract the volume of the conical cavity from the volume of the rectangular solid:
\[ \text{Volume of remaining solid} = \text{Volume of rectangular solid} - \text{Volume of conical cavity} \]
\[ \text{Volume of remaining solid} = 25200 \, \text{cm}^3 - 1232 \, \text{cm}^3 \]
\[ \text{Volume of remaining solid} = 23968 \, \text{cm}^3 \]

### Final Answer:
The volume of the remaining solid is \( 23968 \, \text{cm}^3 \).

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