A solid cylinder whose radius of the base is 12 cm and the height is 8 cm is melted and made into a solid sphere . The radius of the sphere is ________ cm.

Options

A. `6 root (3) 4`
B. 6
C. `8 root (3) 3`
D. `12 root (3) 3`

Answer

To find the radius of the sphere formed by melting a solid cylinder, we need to follow these steps:

### Step 1: Calculate the volume of the cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height.

Given:
- Radius of the cylinder \( r = 12 \) cm
- Height of the cylinder \( h = 8 \) cm

Substituting the values:
\[ V = \pi (12)^2 (8) \]
\[ V = \pi (144) (8) \]
\[ V = 1152\pi \, \text{cm}^3 \]

### Step 2: Set the volume of the sphere equal to the volume of the cylinder
When the cylinder is melted, the volume of the resulting sphere will be equal to the volume of the cylinder. The formula for the volume \( V \) of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]

Setting the volumes equal:
\[ \frac{4}{3} \pi r^3 = 1152\pi \]

### Step 3: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ \frac{4}{3} r^3 = 1152 \]

### Step 4: Solve for \( r^3 \)
Multiply both sides by \( \frac{3}{4} \):
\[ r^3 = 1152 \times \frac{3}{4} \]
\[ r^3 = 864 \]

### Step 5: Calculate the radius \( r \)
To find \( r \), take the cube root of both sides:
\[ r = \sqrt[3]{864} \]

Calculating \( \sqrt[3]{864} \):
\[ r = 9.54 \, \text{cm} \, (\text{approximately}) \]

Thus, the radius of the sphere is approximately **9.54 cm**.

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