A solid metalli sphere of radius 6 cm is melted and made into a solid cylinder of height 32 cm. Find the :
(i) radius of the cylinder
(ii) curved surface area of the cylinder. (Take `pi=3.1`)

Answer

To solve the problem step by step, we will follow the given information and apply the formulas for the volumes and curved surface area.

### Step 1: Find the Volume of the 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:
- Radius of the sphere \( r_1 = 6 \) cm
- Using \( \pi = 3.1 \)

Substituting the values:
\[ V = \frac{4}{3} \times 3.1 \times (6)^3 \]
Calculating \( 6^3 \):
\[ 6^3 = 216 \]
Now substituting this back into the volume formula:
\[ V = \frac{4}{3} \times 3.1 \times 216 \]
Calculating \( \frac{4}{3} \times 3.1 \):
\[ \frac{4 \times 3.1}{3} = \frac{12.4}{3} = 4.1333 \text{ (approximately)} \]
Now multiplying by 216:
\[ V \approx 4.1333 \times 216 \approx 893.3333 \text{ cm}^3 \]

### Step 2: Find the Volume of the Cylinder
The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

Given:
- Height of the cylinder \( h_2 = 32 \) cm
- We need to find the radius \( r_2 \)

Since the volume of the sphere is equal to the volume of the cylinder:
\[ \frac{4}{3} \pi r_1^3 = \pi r_2^2 h_2 \]
Cancelling \( \pi \) from both sides:
\[ \frac{4}{3} r_1^3 = r_2^2 h_2 \]

Substituting the known values:
\[ \frac{4}{3} (6)^3 = r_2^2 (32) \]
Calculating \( \frac{4}{3} \times 216 \):
\[ \frac{864}{3} = 288 \]
So we have:
\[ 288 = r_2^2 \times 32 \]
Dividing both sides by 32:
\[ r_2^2 = \frac{288}{32} = 9 \]
Taking the square root:
\[ r_2 = 3 \text{ cm} \]

### Step 3: Find the Curved Surface Area of the Cylinder
The curved surface area (CSA) of a cylinder is given by:
\[ CSA = 2 \pi r h \]
Substituting the values:
\[ CSA = 2 \times 3.1 \times r_2 \times h_2 \]
Using \( r_2 = 3 \) cm and \( h_2 = 32 \) cm:
\[ CSA = 2 \times 3.1 \times 3 \times 32 \]
Calculating:
\[ CSA = 2 \times 3.1 \times 96 \]
Calculating \( 2 \times 3.1 = 6.2 \):
\[ CSA = 6.2 \times 96 = 595.2 \text{ cm}^2 \]

### Final Answers
(i) Radius of the cylinder \( r_2 = 3 \) cm
(ii) Curved surface area of the cylinder \( CSA = 595.2 \) cm²