Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter `2` `cm` and height `16` `cm` . The diameter of each sphere is .

Options

A. 4 cm
B. 3 cm
C. 2 cm
D. 6 cm

Explanation

To find the diameter of each sphere formed by melting a solid metallic cylinder, we can 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.

Given:
- Diameter of the cylinder = 2 cm, so the radius \( r = \frac{2}{2} = 1 \) cm.
- Height of the cylinder \( h = 16 \) cm.

Now substituting the values:
\[ V = \pi (1)^2 (16) = 16\pi \text{ cm}^3 \]

### Step 2: Calculate the Volume of One Sphere
The volume \( V_s \) of a sphere is given by the formula:
\[ V_s = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

Let the radius of each sphere be \( r_s \). Therefore, the volume of 12 spheres is:
\[ V_{12} = 12 \times V_s = 12 \times \frac{4}{3} \pi r_s^3 = 16\pi \text{ cm}^3 \]

### Step 3: Set the Volumes Equal
Since the volume of the cylinder is equal to the total volume of the spheres, we have:
\[ 16\pi = 12 \times \frac{4}{3} \pi r_s^3 \]

### Step 4: Simplify the Equation
We can cancel \( \pi \) from both sides:
\[ 16 = 12 \times \frac{4}{3} r_s^3 \]

Now, simplify the right side:
\[ 16 = 16 r_s^3 \]

### Step 5: Solve for \( r_s^3 \)
Dividing both sides by 16:
\[ 1 = r_s^3 \]

Taking the cube root of both sides:
\[ r_s = 1 \text{ cm} \]

### Step 6: Calculate the Diameter of Each Sphere
The diameter \( d \) of a sphere is given by:
\[ d = 2r_s \]
Substituting the value of \( r_s \):
\[ d = 2 \times 1 = 2 \text{ cm} \]

### Final Answer
The diameter of each sphere is **2 cm**.

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