If a solid piece of iron in the form of a cuboid of dimensions `49` `cm` `xx` `33` `cm` `xx` `24` `cm`, is moulded to form a solid sphere. Then, radius of the sphere is

Options

A. 21 cm
B. 23 cm
C. 25 cm
D. 19 cm

Explanation

To find the radius of a sphere formed by moulding a solid piece of iron in the shape of a cuboid, we need to 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 the dimensions of the cuboid:
- Length = 49 cm
- Breadth = 33 cm
- Height = 24 cm

Substituting these values into the formula:
\[ V = 49 \, \text{cm} \times 33 \, \text{cm} \times 24 \, \text{cm} \]

### Step 2: Perform the Multiplication
Now, we will calculate the volume:
\[ V = 49 \times 33 \times 24 \]

Calculating step by step:
1. First calculate \( 49 \times 33 \):
\[ 49 \times 33 = 1617 \]
2. Now multiply the result by 24:
\[ 1617 \times 24 = 38808 \, \text{cm}^3 \]

So, the volume of the cuboid is:
\[ V = 38808 \, \text{cm}^3 \]

### Step 3: Set the Volume of the Sphere Equal to the Volume of the Cuboid
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Setting the volume of the cuboid equal to the volume of the sphere:
\[ 38808 = \frac{4}{3} \pi r^3 \]

### Step 4: Solve for \( r^3 \)
Rearranging the equation to solve for \( r^3 \):
\[ r^3 = \frac{38808 \times 3}{4 \pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r^3 = \frac{38808 \times 3}{4 \times \frac{22}{7}} = \frac{38808 \times 3 \times 7}{4 \times 22} \]

### Step 5: Calculate the Right Side
Calculating the numerator:
\[ 38808 \times 3 = 116424 \]
Now calculating:
\[ r^3 = \frac{116424 \times 7}{88} \]
Calculating \(116424 \times 7\):
\[ 116424 \times 7 = 816968 \]
Now divide by 88:
\[ r^3 = \frac{816968}{88} = 9298 \]

### Step 6: Find the Cube Root
Now we need to find \( r \):
\[ r = \sqrt[3]{9298} \]
To simplify, we can factor \( 9298 \):
\[ 9298 = 21 \times 441 \]
Since \( 441 = 21^2 \), we can write:
\[ r^3 = 21^3 \]
Thus:
\[ r = 21 \, \text{cm} \]

### Final Answer
The radius of the sphere is:
\[ \boxed{21 \, \text{cm}} \]