A solid sphere iron ball having radius of 12 cm melted and re- casted into a hollow cylindrical vessel of uniform thickness. If external radius of the base of cylindrical vessel is 10 cm and its height is 64 cm, then find the uniform thickness of the cylindrical vessel?

Options

A. 2 cm
B. 1 cm
C. 2.5 cm
D. 2.25 cm

Answer

To solve the problem step by step, we will follow the given information and apply the formulas for the volumes of a sphere and a hollow cylinder.

### Step 1: Find the volume of the solid sphere
The formula for the volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

Given the radius \( r \) of the sphere is 12 cm, we can substitute this value into the formula:

\[ V = \frac{4}{3} \pi (12)^3 \]

Calculating \( (12)^3 \):

\[ 12^3 = 1728 \]

Now substituting this back into the volume formula:

\[ V = \frac{4}{3} \pi (1728) = \frac{6912}{3} \pi = 2304 \pi \text{ cm}^3 \]

### Step 2: Set up the volume of the hollow cylindrical vessel
The volume \( V \) of a hollow cylinder is given by:

\[ V = \pi h (R^2 - r^2) \]

Where:
- \( R \) is the external radius,
- \( r \) is the internal radius,
- \( h \) is the height.

Given:
- External radius \( R = 10 \) cm,
- Height \( h = 64 \) cm,
- Internal radius \( r \) is what we need to find.

Substituting the known values into the volume formula:

\[ V = \pi (64) (10^2 - r^2) = \pi (64) (100 - r^2) \]

### Step 3: Set the volumes equal
Since the solid sphere is melted and recast into the hollow cylinder, their volumes are equal:

\[ 2304 \pi = \pi (64) (100 - r^2) \]

### Step 4: Cancel \( \pi \) from both sides
Dividing both sides by \( \pi \):

\[ 2304 = 64 (100 - r^2) \]

### Step 5: Solve for \( r^2 \)
Now, divide both sides by 64:

\[ \frac{2304}{64} = 100 - r^2 \]

Calculating \( \frac{2304}{64} \):

\[ \frac{2304}{64} = 36 \]

So we have:

\[ 36 = 100 - r^2 \]

Rearranging gives:

\[ r^2 = 100 - 36 = 64 \]

Taking the square root:

\[ r = 8 \text{ cm} \]

### Step 6: Find the thickness of the cylindrical vessel
The thickness \( t \) of the cylindrical vessel is given by the difference between the external radius and the internal radius:

\[ t = R - r = 10 - 8 = 2 \text{ cm} \]

### Final Answer
The uniform thickness of the cylindrical vessel is:

\[ \boxed{2 \text{ cm}} \]