The radii of the internal and external surfaces of metallic spherical shell are 4 cm and 5 cm respectively. It is mixed and recast into a solid right circular cylinder of height `121/3 cm`.
Then the diameter of the base of the cylinder will be-

Options

A. 3 cm
B. 4 cm
C. 8 cm
D. 6 cm

Answer

To solve the problem step-by-step, we need to find the diameter of a solid right circular cylinder that is formed by recasting a metallic spherical shell.

### Step 1: Calculate the volume of the spherical shell
The volume \( V \) of a hollow sphere (spherical shell) can be calculated using 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 \( r = 4 \) cm

Substituting the values into the formula:
\[ V = \frac{4}{3} \pi (5^3 - 4^3) \]

### Step 2: Calculate \( 5^3 \) and \( 4^3 \)
Calculating the cubes:
\[ 5^3 = 125 \]
\[ 4^3 = 64 \]

### Step 3: Substitute the cubes back into the volume formula
Now substituting back into the volume formula:
\[ V = \frac{4}{3} \pi (125 - 64) \]
\[ V = \frac{4}{3} \pi (61) \]

### Step 4: Calculate the volume of the cylinder
The volume \( V \) of a right circular 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:
- Height \( h = \frac{121}{3} \) cm

### Step 5: Set the volumes equal to each other
Since the volume of the spherical shell is equal to the volume of the cylinder, we can set them equal:
\[ \frac{4}{3} \pi (61) = \pi r^2 \left(\frac{121}{3}\right) \]

### Step 6: Cancel \( \pi \) from both sides
Dividing both sides by \( \pi \):
\[ \frac{4}{3} (61) = r^2 \left(\frac{121}{3}\right) \]

### Step 7: Multiply both sides by 3 to eliminate the fraction
\[ 4 \times 61 = r^2 \times 121 \]
\[ 244 = r^2 \times 121 \]

### Step 8: Solve for \( r^2 \)
Dividing both sides by 121:
\[ r^2 = \frac{244}{121} \]

### Step 9: Calculate \( r \)
Taking the square root of both sides:
\[ r = \sqrt{\frac{244}{121}} = \frac{\sqrt{244}}{11} \]
Calculating \( \sqrt{244} \):
\[ \sqrt{244} \approx 15.62 \quad (\text{since } 244 = 4 \times 61) \]
Thus:
\[ r \approx \frac{15.62}{11} \approx 1.42 \text{ cm} \]

### Step 10: Calculate the diameter of the cylinder
The diameter \( d \) of the cylinder is given by:
\[ d = 2r \]
Substituting the value of \( r \):
\[ d = 2 \times 1.42 \approx 2.84 \text{ cm} \]

### Final Answer
The diameter of the base of the cylinder is approximately **3 cm**.

---