The sum of the inner and the outer curved surfaces of a hollow metallic cylinder is `1056 cm ^(2)` and the volume of material in it is `1056 cm^(3).` Find its internal and external radii. Given that the height of the cylinder is 21 cm.

Answer

To solve the problem step by step, we will use the given information about the hollow metallic cylinder.

### Given:
- Height of the cylinder (h) = 21 cm
- Sum of the inner and outer curved surfaces = 1056 cm²
- Volume of the material = 1056 cm³

### Let:
- Internal radius = r (smaller radius)
- External radius = R (larger radius)

### Step 1: Write the equation for the sum of the curved surface areas
The curved surface area of a hollow cylinder is given by:
- Inner curved surface area = \(2 \pi r h\)
- Outer curved surface area = \(2 \pi R h\)

According to the problem:
\[ 2 \pi r h + 2 \pi R h = 1056 \]

### Step 2: Factor out common terms
Factoring out \(2 \pi h\):
\[ 2 \pi h (r + R) = 1056 \]

### Step 3: Substitute the values of \(h\) and \(\pi\)
Using \(\pi \approx \frac{22}{7}\) and \(h = 21\):
\[ 2 \times \frac{22}{7} \times 21 (r + R) = 1056 \]

### Step 4: Simplify the equation
Calculating \(2 \times \frac{22}{7} \times 21\):
\[ \frac{44 \times 21}{7} = \frac{924}{7} = 132 \]
Thus, we have:
\[ 132 (r + R) = 1056 \]

### Step 5: Solve for \(r + R\)
Dividing both sides by 132:
\[ r + R = \frac{1056}{132} = 8 \]

### Step 6: Write the equation for the volume of the material
The volume of the material in the hollow cylinder is given by:
\[ \text{Volume} = \text{Outer volume} - \text{Inner volume} \]
\[ \pi R^2 h - \pi r^2 h = 1056 \]

### Step 7: Factor out common terms
Factoring out \(\pi h\):
\[ \pi h (R^2 - r^2) = 1056 \]

### Step 8: Substitute the values of \(h\) and \(\pi\)
Using \(\pi \approx \frac{22}{7}\) and \(h = 21\):
\[ \frac{22}{7} \times 21 (R^2 - r^2) = 1056 \]

### Step 9: Simplify the equation
Calculating \(\frac{22 \times 21}{7}\):
\[ \frac{462}{7} = 66 \]
Thus, we have:
\[ 66 (R^2 - r^2) = 1056 \]

### Step 10: Solve for \(R^2 - r^2\)
Dividing both sides by 66:
\[ R^2 - r^2 = \frac{1056}{66} = 16 \]

### Step 11: Use the difference of squares
Using the identity \(R^2 - r^2 = (R + r)(R - r)\):
\[ (R + r)(R - r) = 16 \]

### Step 12: Substitute \(R + r\) from Step 5
We know \(R + r = 8\):
\[ 8(R - r) = 16 \]

### Step 13: Solve for \(R - r\)
Dividing both sides by 8:
\[ R - r = 2 \]

### Step 14: Solve the system of equations
Now we have two equations:
1. \(R + r = 8\)
2. \(R - r = 2\)

Adding both equations:
\[ (R + r) + (R - r) = 8 + 2 \]
\[ 2R = 10 \implies R = 5 \]

### Step 15: Find \(r\)
Substituting \(R = 5\) into \(R + r = 8\):
\[ 5 + r = 8 \implies r = 3 \]

### Final Answer:
- Internal radius \(r = 3\) cm
- External radius \(R = 5\) cm