A solid piece of iron in the form of a cuboid of dimensions (49 `xx` 33cm `xx` 24 cm) is moulded to form a solid sphere . The radius of the sphere is

Options

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

Explanation

To find the radius of the sphere formed from a solid piece of iron in the shape of a cuboid, we need to follow these steps:

### Step-by-Step Solution:

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 of the cuboid are:
- Length \( L = 49 \) cm
- Breadth \( B = 33 \) cm
- Height \( H = 24 \) cm

Plugging in the values:
\[ V = 49 \times 33 \times 24 \]

2. **Calculate the Volume**:
First, calculate \( 49 \times 33 \):
\[ 49 \times 33 = 1617 \]
Now, multiply this result by 24:
\[ 1617 \times 24 = 38808 \text{ cm}^3 \]
So, the volume of the cuboid is \( 38808 \text{ cm}^3 \).

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 sphere equal to the volume of the cuboid:
\[ \frac{4}{3} \pi r^3 = 38808 \]

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

5. **Calculate the Right Side**:
First, calculate \( 38808 \times 3 \):
\[ 38808 \times 3 = 116424 \]
Now calculate \( 4 \times 22 = 88 \):
\[ r^3 = \frac{116424 \times 7}{88} \]
Calculate \( 116424 \div 88 \):
\[ 116424 \div 88 = 1323 \]
Now multiply by 7:
\[ r^3 = 1323 \times 7 = 9261 \]

6. **Find \( r \)**:
Now, take the cube root of \( 9261 \):
\[ r = \sqrt[3]{9261} = 21 \text{ cm} \]

### Final Answer:
The radius of the sphere is \( 21 \) cm.