A solid metallic 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, we will follow these steps:

### Step 1: Calculate the Volume of the Sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r_1^3 \]
where \( r_1 \) is the radius of the sphere. Given \( r_1 = 6 \) cm, we can substitute this value into the formula.

\[ V = \frac{4}{3} \pi (6)^3 \]
Calculating \( (6)^3 \):
\[ 6^3 = 216 \]
Now substituting back:
\[ V = \frac{4}{3} \pi (216) = \frac{864}{3} \pi = 288 \pi \]

### Step 2: Calculate the Volume of the Cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r_2^2 h \]
where \( r_2 \) is the radius of the cylinder and \( h \) is the height. Given \( h = 32 \) cm, we can set the volume of the sphere equal to the volume of the cylinder since the sphere is melted to form the cylinder.

\[ 288 \pi = \pi r_2^2 (32) \]

### Step 3: Simplify the Equation
We can divide both sides by \( \pi \):
\[ 288 = r_2^2 (32) \]
Now, divide both sides by 32 to solve for \( r_2^2 \):
\[ r_2^2 = \frac{288}{32} \]
Calculating \( \frac{288}{32} \):
\[ r_2^2 = 9 \]

### Step 4: Find the Radius of the Cylinder
Taking the square root of both sides:
\[ r_2 = \sqrt{9} = 3 \text{ cm} \]

### Step 5: Calculate the Curved Surface Area of the Cylinder
The curved surface area \( A \) of a cylinder is given by the formula:
\[ A = 2 \pi r_2 h \]
Substituting \( r_2 = 3 \) cm and \( h = 32 \) cm:
\[ A = 2 \pi (3)(32) \]
Calculating:
\[ A = 6 \pi (32) = 192 \pi \]
Now substituting \( \pi = 3.1 \):
\[ A = 192 \times 3.1 = 595.2 \text{ cm}^2 \]

### Final Answers:
(i) The radius of the cylinder is \( 3 \) cm.
(ii) The curved surface area of the cylinder is \( 595.2 \) cm².