The difference between the outer curved surface area and the inner curved surface area of a hollow cylinder is `352 cm^(2).` If its height is 28 cm and the volume of material in it is `704 cm^(3),` find its external curved surface area.

Answer

To solve the problem step by step, we will use the information provided in the question about the hollow cylinder.

### Step 1: Define Variables
Let:
- \( R \) = external radius of the hollow cylinder
- \( r \) = internal radius of the hollow cylinder
- \( H \) = height of the cylinder = 28 cm

### Step 2: Set Up the Equation for the Difference in Curved Surface Areas
The difference between the outer curved surface area and the inner curved surface area is given as \( 352 \, \text{cm}^2 \). The formulas for the curved surface areas are:
- External curved surface area = \( 2 \pi R H \)
- Internal curved surface area = \( 2 \pi r H \)

Thus, we can write:
\[ 2 \pi R H - 2 \pi r H = 352 \]
Factoring out \( 2 \pi H \):
\[ 2 \pi H (R - r) = 352 \]
Substituting \( H = 28 \):
\[ 2 \pi (28) (R - r) = 352 \]

### Step 3: Solve for \( R - r \)
Now, substituting \( \pi \) as \( \frac{22}{7} \):
\[ 2 \times \frac{22}{7} \times 28 (R - r) = 352 \]
Calculating the left side:
\[ \frac{44 \times 28}{7} (R - r) = 352 \]
Calculating \( \frac{44 \times 28}{7} \):
\[ \frac{1232}{7} (R - r) = 352 \]
Now, multiplying both sides by \( 7 \):
\[ 1232 (R - r) = 2464 \]
Dividing both sides by \( 1232 \):
\[ R - r = \frac{2464}{1232} = 2 \]

### Step 4: Set Up the Equation for the Volume of Material
The volume of the material in the hollow cylinder is given as \( 704 \, \text{cm}^3 \). The volume of the hollow cylinder can be expressed as:
\[ \text{Volume} = \text{External Volume} - \text{Internal Volume} \]
\[ \pi R^2 H - \pi r^2 H = 704 \]
Factoring out \( \pi H \):
\[ \pi H (R^2 - r^2) = 704 \]
Substituting \( H = 28 \):
\[ \pi (28) (R^2 - r^2) = 704 \]
Substituting \( \pi \) as \( \frac{22}{7} \):
\[ \frac{22}{7} \times 28 (R^2 - r^2) = 704 \]
Calculating \( \frac{22 \times 28}{7} \):
\[ \frac{616}{7} (R^2 - r^2) = 704 \]
Multiplying both sides by \( 7 \):
\[ 616 (R^2 - r^2) = 4928 \]
Dividing both sides by \( 616 \):
\[ R^2 - r^2 = \frac{4928}{616} = 8 \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( R - r = 2 \)
2. \( R^2 - r^2 = 8 \)

Using the identity \( R^2 - r^2 = (R - r)(R + r) \):
\[ 8 = (2)(R + r) \]
Thus:
\[ R + r = \frac{8}{2} = 4 \]

### Step 6: Solve for \( R \) and \( r \)
Now we have the system of equations:
1. \( R - r = 2 \)
2. \( R + r = 4 \)

Adding these two equations:
\[ (R - r) + (R + r) = 2 + 4 \]
\[ 2R = 6 \implies R = 3 \]
Substituting \( R = 3 \) into \( R + r = 4 \):
\[ 3 + r = 4 \implies r = 1 \]

### Step 7: Calculate the External Curved Surface Area
Now we can find the external curved surface area:
\[ \text{External Curved Surface Area} = 2 \pi R H = 2 \pi (3)(28) \]
Substituting \( \pi = \frac{22}{7} \):
\[ = 2 \times \frac{22}{7} \times 3 \times 28 \]
Calculating:
\[ = \frac{132 \times 28}{7} = \frac{3696}{7} = 528 \, \text{cm}^2 \]

### Final Answer
The external curved surface area is \( 528 \, \text{cm}^2 \).

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