A hollow sphere of internal and external diameters 4 cm and 8 cm respectively is melted to form a solid cylinder of base diameter 8 cm. The height of the cylinder is approximately :

Options

A. 4.5 cm
B. 4.57 cm
C. 4.67 cm
D. 4.7 cm

Answer

To solve the problem, we need to find the height of a solid cylinder formed by melting a hollow sphere. Let's break it down step by step.

### Step 1: Identify the given dimensions
- Internal diameter of the hollow sphere = 4 cm
- External diameter of the hollow sphere = 8 cm

### Step 2: Calculate the radii
- Internal radius (r) = Internal diameter / 2 = 4 cm / 2 = 2 cm
- External radius (R) = External diameter / 2 = 8 cm / 2 = 4 cm

### Step 3: Calculate the volume of the hollow sphere
The volume of the hollow sphere is given by the formula:
\[ \text{Volume} = \frac{4}{3} \pi (R^3 - r^3) \]
Substituting the values of R and r:
\[ \text{Volume} = \frac{4}{3} \pi (4^3 - 2^3) \]
Calculating the cubes:
\[ = \frac{4}{3} \pi (64 - 8) = \frac{4}{3} \pi (56) \]

### Step 4: Calculate the volume of the cylinder
The volume of the cylinder is given by the formula:
\[ \text{Volume} = \pi r^2 h \]
Where r is the radius of the cylinder's base. Since the diameter of the cylinder is also 8 cm, the radius is:
\[ r = \frac{8}{2} = 4 \text{ cm} \]
Thus, the volume of the cylinder can be expressed as:
\[ \text{Volume} = \pi (4^2) h = \pi (16) h \]

### Step 5: Set the volumes equal to each other
Since the hollow sphere is melted to form the cylinder, their volumes are equal:
\[ \frac{4}{3} \pi (56) = \pi (16) h \]

### Step 6: Cancel out \(\pi\) from both sides
\[ \frac{4}{3} (56) = 16h \]

### Step 7: Solve for h
First, calculate the left side:
\[ \frac{4 \times 56}{3} = \frac{224}{3} \]
Now, set it equal to \(16h\):
\[ \frac{224}{3} = 16h \]
Now, isolate h:
\[ h = \frac{224}{3 \times 16} \]
Calculating \(3 \times 16 = 48\):
\[ h = \frac{224}{48} \]
Now simplify:
\[ h = \frac{224 \div 16}{48 \div 16} = \frac{14}{3} \approx 4.67 \text{ cm} \]

### Final Answer
The height of the cylinder is approximately \(4.67 \text{ cm}\).

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