How many spherical balls each of 2cm radius can be made out of a solid metallic cube of edge 44 cm? (Take` pi = (22)/(7)`)

Options

A. 2420
B. 2532
C. 2448
D. 2541

Answer

To determine how many spherical balls of radius 2 cm can be made from a solid metallic cube with an edge of 44 cm, 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 an edge of the cube. Here, \( a = 44 \) cm.

Calculating the volume:
\[ V = 44^3 = 44 \times 44 \times 44 \]
Calculating \( 44 \times 44 = 1936 \), then:
\[ V = 1936 \times 44 = 85384 \text{ cm}^3 \]

### Step 2: Calculate the Volume of One Sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere. Here, \( r = 2 \) cm and \( \pi = \frac{22}{7} \).

Calculating the volume:
\[ V = \frac{4}{3} \times \frac{22}{7} \times (2)^3 \]
Calculating \( (2)^3 = 8 \):
\[ V = \frac{4}{3} \times \frac{22}{7} \times 8 \]
Now, simplifying:
\[ V = \frac{4 \times 22 \times 8}{3 \times 7} = \frac{704}{21} \text{ cm}^3 \]

### Step 3: Calculate the Number of Spheres
To find the number of spheres that can be made, we divide the volume of the cube by the volume of one sphere:
\[ \text{Number of spheres} = \frac{\text{Volume of cube}}{\text{Volume of sphere}} = \frac{85384}{\frac{704}{21}} \]
This can be simplified as:
\[ \text{Number of spheres} = 85384 \times \frac{21}{704} \]

### Step 4: Simplifying the Calculation
Calculating \( 85384 \div 704 \):
\[ 85384 \div 704 = 121 \]
Now, multiplying by 21:
\[ \text{Number of spheres} = 121 \times 21 = 2541 \]

### Final Answer
Thus, the number of spherical balls that can be made is:
\[ \boxed{2541} \]