The radii of internal and external surfaces of a hollow spherical shell are 3 cm and 5 cm respectively. It is melted and recast into a solid cylinder of diameter 14 cm. Find the height of the cylinder.

Answer

To solve the problem step by step, we need to find the height of the cylinder formed by melting the hollow spherical shell. Here’s how we can do it:

### Step 1: Calculate the volume of the hollow spherical shell
The volume \( V \) of a hollow sphere is given by the formula:
\[ V = \frac{4}{3} \pi (R^3 - r^3) \]
where \( R \) is the external radius and \( r \) is the internal radius.

Given:
- External radius \( R = 5 \) cm
- Internal radius \( r = 3 \) cm

Substituting the values:
\[ V = \frac{4}{3} \pi (5^3 - 3^3) \]
Calculating \( 5^3 \) and \( 3^3 \):
\[ 5^3 = 125 \quad \text{and} \quad 3^3 = 27 \]
Now substituting these values:
\[ V = \frac{4}{3} \pi (125 - 27) = \frac{4}{3} \pi (98) \]
Thus, the volume of the hollow spherical shell is:
\[ V = \frac{392}{3} \pi \text{ cm}^3 \]

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

Given the diameter of the cylinder is 14 cm, we can find the radius:
\[ r = \frac{d}{2} = \frac{14}{2} = 7 \text{ cm} \]

### Step 3: Set the volumes equal to each other
Since the hollow spherical shell is melted and recast into the cylinder, the volumes are equal:
\[ \frac{392}{3} \pi = \pi (7^2) h \]
Cancelling \( \pi \) from both sides:
\[ \frac{392}{3} = 49h \]

### Step 4: Solve for the height \( h \)
To find \( h \), we rearrange the equation:
\[ h = \frac{392}{3 \times 49} \]
Calculating \( 3 \times 49 = 147 \):
\[ h = \frac{392}{147} \]
Now simplifying:
\[ h = \frac{392 \div 49}{147 \div 49} = \frac{8}{3} \text{ cm} \]

### Final Answer
The height of the cylinder is:
\[ h = \frac{8}{3} \text{ cm} \approx 2.67 \text{ cm} \]

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