The total surface area of a solid cylinder is `616 cm^(2)`. If the ratio between its curved surface area and total surface area is 1:2, find the volume of the cylinder.

Answer

To find the volume of the cylinder given the total surface area and the ratio of the curved surface area to the total surface area, follow these steps:

### Step 1: Understand the given information
- Total Surface Area (TSA) of the cylinder = 616 cm²
- Ratio of Curved Surface Area (CSA) to Total Surface Area = 1:2

### Step 2: Write the formulas
- The formula for the Total Surface Area (TSA) of a cylinder is:
\[ \text{TSA} = 2\pi r (r + h) \]
- The formula for the Curved Surface Area (CSA) of a cylinder is:
\[ \text{CSA} = 2\pi rh \]

### Step 3: Set up the ratio
From the given ratio of CSA to TSA:
\[ \frac{\text{CSA}}{\text{TSA}} = \frac{1}{2} \]
Substituting the formulas:
\[ \frac{2\pi rh}{2\pi r(r + h)} = \frac{1}{2} \]
This simplifies to:
\[ \frac{h}{r + h} = \frac{1}{2} \]

### Step 4: Cross-multiply to solve for h
Cross-multiplying gives:
\[ 2h = r + h \]
Rearranging the equation:
\[ 2h - h = r \implies h = r \]

### Step 5: Substitute h in the TSA formula
Since \( h = r \), we can substitute \( h \) in the TSA formula:
\[ \text{TSA} = 2\pi r (r + r) = 2\pi r (2r) = 4\pi r^2 \]
Setting this equal to the given TSA:
\[ 4\pi r^2 = 616 \]

### Step 6: Solve for r
Rearranging gives:
\[ \pi r^2 = \frac{616}{4} = 154 \]
Substituting \( \pi \) as \( \frac{22}{7} \):
\[ \frac{22}{7} r^2 = 154 \]
Multiplying both sides by 7:
\[ 22r^2 = 154 \times 7 \]
Calculating \( 154 \times 7 \):
\[ 154 \times 7 = 1078 \]
So:
\[ 22r^2 = 1078 \]
Dividing both sides by 22:
\[ r^2 = \frac{1078}{22} = 49 \]
Taking the square root:
\[ r = 7 \text{ cm} \]

### Step 7: Find h
Since \( h = r \):
\[ h = 7 \text{ cm} \]

### Step 8: Calculate the volume of the cylinder
The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
Substituting the values:
\[ V = \frac{22}{7} \times 7^2 \times 7 \]
Calculating:
\[ V = \frac{22}{7} \times 49 \times 7 = 22 \times 49 = 1078 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is \( 1078 \text{ cm}^3 \).

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