The internal radius and thickness of a hollow metallic pipe are 24 cm and 1 cm respectively. If it is melted and recast into a solid cylinder of equal length, then find the diameter of the solid cylinder.

Options

A. 7 cm
B. 14 cm
C. 96 cm
D. 98 cm

Answer

To solve the problem, we need to find the diameter of a solid cylinder that can be formed by melting a hollow metallic pipe. Let's break down the steps:

### Step 1: Determine the dimensions of the hollow pipe
- **Internal radius (r)** = 24 cm
- **Thickness (t)** = 1 cm
- **External radius (R)** = Internal radius + Thickness = 24 cm + 1 cm = 25 cm

### Step 2: Calculate the volume of the hollow pipe
The volume \( V \) of the hollow cylinder (pipe) can be calculated using the formula:
\[ V = \pi h (R^2 - r^2) \]
where \( h \) is the height (or length) of the cylinder. Since the height is not given, we will keep it as \( h \).

Substituting the values:
\[ V = \pi h (25^2 - 24^2) \]
Calculating \( 25^2 \) and \( 24^2 \):
\[ 25^2 = 625 \quad \text{and} \quad 24^2 = 576 \]
Thus,
\[ V = \pi h (625 - 576) = \pi h (49) = 49\pi h \]

### Step 3: Set up the volume of the solid cylinder
Let the radius of the solid cylinder be \( R_s \). The volume \( V_s \) of the solid cylinder is given by:
\[ V_s = \pi R_s^2 h \]

### Step 4: Equate the volumes
Since the hollow pipe is melted and recast into the solid cylinder, the volumes are equal:
\[ 49\pi h = \pi R_s^2 h \]
We can cancel \( \pi h \) from both sides (assuming \( h \neq 0 \)):
\[ 49 = R_s^2 \]

### Step 5: Solve for \( R_s \)
Taking the square root of both sides:
\[ R_s = \sqrt{49} = 7 \text{ cm} \]

### Step 6: Find the diameter of the solid cylinder
The diameter \( D \) of the solid cylinder is given by:
\[ D = 2R_s = 2 \times 7 = 14 \text{ cm} \]

### Final Answer
The diameter of the solid cylinder is **14 cm**.

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