A metallic solid cuboid of sides 44 cm, 32 cm and 36 cm melted and converted into some number of spheres of radius 12 cm. How many such sphere can be made with the metal `(pi = 22//7)` ?

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem of how many spheres can be made from a metallic solid cuboid, we will follow these steps:

### Step 1: Calculate the Volume of the Cuboid
The volume \( V \) of a cuboid is given by the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
Given dimensions:
- Length = 44 cm
- Breadth = 32 cm
- Height = 36 cm

Substituting the values:
\[ V = 44 \times 32 \times 36 \]

### Step 2: Calculate the Volume
Now, let's perform the multiplication:
\[ 44 \times 32 = 1408 \]
\[ 1408 \times 36 = 50688 \text{ cm}^3 \]
Thus, the volume of the cuboid is \( 50688 \text{ cm}^3 \).

### Step 3: Calculate the Volume of One Sphere
The volume \( V_s \) of a sphere is given by the formula:
\[ V_s = \frac{4}{3} \pi r^3 \]
Given radius \( r = 12 \) cm and using \( \pi = \frac{22}{7} \):
\[ V_s = \frac{4}{3} \times \frac{22}{7} \times (12)^3 \]

### Step 4: Calculate \( 12^3 \)
First, calculate \( 12^3 \):
\[ 12^3 = 12 \times 12 \times 12 = 1728 \]

### Step 5: Substitute and Calculate the Volume of One Sphere
Now substitute \( 12^3 \) into the volume formula:
\[ V_s = \frac{4}{3} \times \frac{22}{7} \times 1728 \]

### Step 6: Simplify the Volume of One Sphere
Calculating \( \frac{4}{3} \times 1728 \):
\[ \frac{4 \times 1728}{3} = \frac{6912}{3} = 2304 \]
Now, substitute this back:
\[ V_s = \frac{22}{7} \times 2304 \]
Calculating \( \frac{22 \times 2304}{7} \):
\[ 22 \times 2304 = 50688 \]
Now divide by 7:
\[ V_s = \frac{50688}{7} = 7240.57 \text{ cm}^3 \text{ (approximately)} \]

### Step 7: Calculate the Number of Spheres
To find the number of spheres \( n \), we divide the volume of the cuboid by the volume of one sphere:
\[ n = \frac{V_{\text{cuboid}}}{V_s} = \frac{50688}{7240.57} \]
Calculating this gives:
\[ n \approx 7 \]

### Final Answer
Thus, the number of spheres that can be made is \( \boxed{7} \).