A solid piece of iron of dimensions 49 cm x 33 cm x 24 cm is moulded into a sphere. Find the radius of the sphere.

Answer

To find the radius of the sphere that can be formed from a solid piece of iron with dimensions 49 cm x 33 cm x 24 cm, we will follow these steps:

### Step 1: Calculate the Volume of the Cuboid
The volume \( V \) of a cuboid is given by the formula:
\[ V = \text{Length} \times \text{Breadth} \times \text{Height} \]
Substituting the given dimensions:
\[ V = 49 \, \text{cm} \times 33 \, \text{cm} \times 24 \, \text{cm} \]

### Step 2: Perform the Multiplication
Calculating 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 this result by 24:
\[ 1617 \times 24 = 38808 \, \text{cm}^3 \]

### 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:
\[ V = \frac{4}{3} \pi r^3 \]
Since the volume of the iron piece (cuboid) is equal to the volume of the sphere, we have:
\[ 38808 = \frac{4}{3} \pi r^3 \]

### 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: Simplify the Expression
Calculating the right-hand side:
1. Calculate \( 4 \times 22 = 88 \).
2. Now calculate:
\[ r^3 = \frac{38808 \times 21}{88} \]
3. Simplifying \( \frac{38808}{88} \):
\[ 38808 \div 88 = 441 \]
4. Therefore:
\[ r^3 = 441 \times 21 = 9261 \]

### Step 6: Find the Radius \( r \)
To find \( r \), we take the cube root of \( 9261 \):
\[ r = \sqrt[3]{9261} \]
Calculating the cube root:
\[ r = 21 \, \text{cm} \]

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