A solid rectangular block of metal 49 cm by 44 cm by 18 cm is melted and formed into a solid sphere. Calculate the radius of the sphere.

Answer

To find the radius of a sphere formed by melting a solid rectangular block of metal, we will follow these steps:

### Step 1: Calculate the Volume of the Rectangular Block
The volume \( V \) of a rectangular block is given by the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
Given dimensions are:
- Length = 49 cm
- Breadth = 44 cm
- Height = 18 cm

Substituting the values:
\[ V = 49 \times 44 \times 18 \]

### Step 2: Perform the Multiplication
First, we calculate \( 49 \times 18 \):
\[ 49 \times 18 = 882 \]

Now, multiply the result by 44:
\[ 882 \times 44 \]
Calculating this:
\[ 882 \times 44 = 38808 \text{ cm}^3 \]
So, the volume of the rectangular block is \( 38808 \text{ cm}^3 \).

### Step 3: Set the Volume of the Sphere Equal to the Volume of the Block
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 block:
\[ \frac{4}{3} \pi r^3 = 38808 \]

### Step 4: Solve for \( r^3 \)
To isolate \( r^3 \), we first multiply both sides by \( \frac{3}{4} \):
\[ \pi r^3 = \frac{3 \times 38808}{4} \]
Calculating the right side:
\[ \frac{3 \times 38808}{4} = 29106 \]
So, we have:
\[ \pi r^3 = 29106 \]

Now, divide both sides by \( \pi \) (using \( \pi \approx \frac{22}{7} \)):
\[ r^3 = \frac{29106 \times 7}{22} \]
Calculating:
\[ r^3 = \frac{204742}{22} = 9301 \]

### Step 5: Find the Cube Root of \( r^3 \)
To find \( r \), we need to calculate the cube root of \( 9301 \):
\[ r = \sqrt[3]{9301} \]

### Step 6: Calculate the Cube Root
Using prime factorization or estimation, we find:
\[ r = 21 \text{ cm} \]

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

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