The radius of cross-section of a solid cylindrical rod of iron is 50 cm. The cylinder is melted down and formed into 6 solid spherical balls of the same radius as that of the cylinder. The length of the rod (in metres) is

Options

A. `0.8`
B. 2
C. 3
D. 4

Answer

To find the length of the cylindrical rod, we will follow these steps:

### Step 1: Calculate the Volume of the Cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height (or length) of the cylinder. Given that the radius \( r = 50 \) cm, we can express the volume as:
\[ V = \pi (50)^2 h = \pi \times 2500 \times h \]

### Step 2: Calculate the Volume of the Spheres
The volume \( V \) of one sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Since we have 6 spheres, the total volume of the spheres is:
\[ V = 6 \times \frac{4}{3} \pi r^3 = 8 \pi r^3 \]
Substituting \( r = 50 \) cm:
\[ V = 8 \pi (50)^3 = 8 \pi \times 125000 = 1000000 \pi \]

### Step 3: Set the Volumes Equal
Since the volume of the melted cylinder is equal to the total volume of the spheres, we can set the two equations equal to each other:
\[ \pi \times 2500 \times h = 1000000 \pi \]

### Step 4: Simplify the Equation
We can cancel \( \pi \) from both sides:
\[ 2500h = 1000000 \]

### Step 5: Solve for \( h \)
Now, divide both sides by 2500 to find \( h \):
\[ h = \frac{1000000}{2500} = 400 \text{ cm} \]

### Step 6: Convert to Meters
Since the question asks for the length in meters, we convert centimeters to meters:
\[ h = \frac{400}{100} = 4 \text{ meters} \]

### Final Answer
The length of the rod is \( 4 \) meters.

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