A solid sphere of radius 6 cm is melted into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is 5 cm and its height is 32 cm, find the uniform thickness of the cylinder.

Options

A. 2 cm
B. 3 cm
C. 1 cm
D. 3.5 cm

Answer

To solve the problem, we need to find the uniform thickness of the hollow cylinder formed by melting a solid sphere. Let's break down the solution step by step.

### Step 1: Calculate the Volume of the Solid Sphere
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Where \( r \) is the radius of the sphere. Given that the radius of the sphere is 6 cm, we can substitute this value into the formula:
\[ V = \frac{4}{3} \pi (6)^3 \]

### Step 2: Compute \( (6)^3 \)
Calculating \( (6)^3 \):
\[ (6)^3 = 216 \]
Now substituting back into the volume formula:
\[ V = \frac{4}{3} \pi (216) \]

### Step 3: Calculate the Volume of the Sphere
Now we calculate the volume:
\[ V = \frac{4 \times 216}{3} \pi = 288 \pi \text{ cm}^3 \]

### Step 4: Set Up the Volume of the Hollow Cylinder
The volume of a hollow cylinder can be calculated using the formula:
\[ V = \pi h (R^2 - r^2) \]
Where:
- \( R \) is the external radius of the cylinder (5 cm),
- \( r \) is the internal radius,
- \( h \) is the height of the cylinder (32 cm).

### Step 5: Substitute Known Values into the Cylinder Volume Formula
Substituting the known values into the volume formula:
\[ 288 \pi = \pi (32) (5^2 - r^2) \]
Simplifying this gives:
\[ 288 = 32 (25 - r^2) \]

### Step 6: Solve for \( r^2 \)
Dividing both sides by 32:
\[ \frac{288}{32} = 25 - r^2 \]
Calculating \( \frac{288}{32} \):
\[ 9 = 25 - r^2 \]
Now, rearranging gives:
\[ r^2 = 25 - 9 = 16 \]

### Step 7: Calculate the Internal Radius \( r \)
Taking the square root of both sides:
\[ r = \sqrt{16} = 4 \text{ cm} \]

### Step 8: Find the Thickness of the Cylinder
The thickness \( t \) of the cylinder is the difference between the external radius \( R \) and the internal radius \( r \):
\[ t = R - r = 5 - 4 = 1 \text{ cm} \]

### Final Answer
The uniform thickness of the hollow cylinder is **1 cm**.

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