A solid metallic cuboid of dimension 24 cm `xx 11 cm xx 7` cm is melted and recast into solid cones of base radius 3.5 cm and height 6 cm. Find the number of cones so formed.

Answer

To solve the problem, we need to find the number of solid cones that can be formed from a solid metallic cuboid by following these steps:

### Step 1: Calculate the Volume of the Cuboid
The volume \( V \) of a cuboid is given by the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
For the given dimensions of the cuboid:
- Length = 24 cm
- Width = 11 cm
- Height = 7 cm

Substituting the values:
\[ V = 24 \, \text{cm} \times 11 \, \text{cm} \times 7 \, \text{cm} \]
\[ V = 1848 \, \text{cm}^3 \]

### Step 2: Calculate the Volume of One Cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( r \) is the base radius
- \( h \) is the height

For the given cone:
- Base radius \( r = 3.5 \, \text{cm} \)
- Height \( h = 6 \, \text{cm} \)

Substituting the values:
\[ V = \frac{1}{3} \pi (3.5 \, \text{cm})^2 (6 \, \text{cm}) \]
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \, \text{cm}^2 \]
Now substituting back:
\[ V = \frac{1}{3} \pi (12.25 \, \text{cm}^2) (6 \, \text{cm}) \]
\[ V = \frac{1}{3} \pi (73.5 \, \text{cm}^3) \]
\[ V = 24.5 \pi \, \text{cm}^3 \]

### Step 3: Calculate the Number of Cones Formed
To find the number of cones formed, we divide the volume of the cuboid by the volume of one cone:
\[ \text{Number of cones} = \frac{\text{Volume of cuboid}}{\text{Volume of one cone}} \]
Substituting the volumes:
\[ \text{Number of cones} = \frac{1848 \, \text{cm}^3}{24.5 \pi \, \text{cm}^3} \]
Using \( \pi \approx 3.14 \):
\[ \text{Number of cones} = \frac{1848}{24.5 \times 3.14} \]
Calculating \( 24.5 \times 3.14 \):
\[ 24.5 \times 3.14 \approx 76.83 \]
Now substituting back:
\[ \text{Number of cones} = \frac{1848}{76.83} \approx 24.06 \]
Since the number of cones must be a whole number, we take the integer part:
\[ \text{Number of cones} = 24 \]

### Final Answer:
The number of cones formed is **24**.

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