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 solve the problem step by step, we will use the given information about the total surface area and the ratio of the curved surface area to the total surface area of the cylinder.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- Total Surface Area (TSA) of the cylinder = 616 cm²
- The ratio of Curved Surface Area (CSA) to Total Surface Area (TSA) = 1:2

2. **Formulas**:
- The formula for the Total Surface Area of a cylinder is:
\[ \text{TSA} = 2\pi rh + 2\pi r^2 \]
- The formula for the Curved Surface Area of a cylinder is:
\[ \text{CSA} = 2\pi rh \]

3. **Setting Up the Ratio**:
- Since the ratio of CSA to TSA is 1:2, we can express this as:
\[ \frac{\text{CSA}}{\text{TSA}} = \frac{1}{2} \]
- This implies:
\[ \text{CSA} = \frac{1}{2} \times \text{TSA} \]

4. **Substituting the Values**:
- From the TSA, we have:
\[ \text{CSA} = \frac{1}{2} \times 616 = 308 \text{ cm}^2 \]

5. **Using the CSA Formula**:
- Now, using the formula for CSA:
\[ 2\pi rh = 308 \]

6. **Expressing h in terms of r**:
- From the ratio, we know that:
\[ \text{TSA} = \text{CSA} + 2\pi r^2 \]
- Substituting the values:
\[ 616 = 308 + 2\pi r^2 \]
- Simplifying gives:
\[ 2\pi r^2 = 616 - 308 = 308 \]

7. **Solving for r**:
- Now, substituting \(\pi \approx \frac{22}{7}\):
\[ 2 \times \frac{22}{7} r^2 = 308 \]
- Multiplying both sides by 7:
\[ 44r^2 = 308 \times 7 \]
- Calculating \(308 \times 7 = 2156\):
\[ 44r^2 = 2156 \]
- Dividing by 44:
\[ r^2 = \frac{2156}{44} = 49 \]
- Taking the square root:
\[ r = 7 \text{ cm} \]

8. **Finding h**:
- Now substituting \(r\) back into the CSA formula:
\[ 2\pi rh = 308 \]
- Substituting \(\pi\) and \(r\):
\[ 2 \times \frac{22}{7} \times 7 \times h = 308 \]
- Simplifying gives:
\[ 44h = 308 \]
- Dividing by 44:
\[ h = \frac{308}{44} = 7 \text{ cm} \]

9. **Calculating the Volume**:
- The formula for the volume of a cylinder is:
\[ V = \pi r^2 h \]
- Substituting the values we found:
\[ V = \frac{22}{7} \times 7^2 \times 7 \]
- This simplifies to:
\[ V = \frac{22}{7} \times 49 \times 7 = 22 \times 49 = 1078 \text{ cm}^3 \]

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