The curved surface area of a cylinder is 264 `m^(2)` and its volume is 924 `m^(3)`. Find the ratio of its height to its diameter.

Answer

To solve the problem, we need to find the ratio of the height to the diameter of a cylinder given its curved surface area (CSA) and volume.

### Step-by-Step Solution:

1. **Understand the formulas**:
- The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi rh \]
- The formula for the volume (V) of a cylinder is given by:
\[ V = \pi r^2 h \]

2. **Substitute the given values**:
- From the problem, we have:
\[ 2\pi rh = 264 \quad \text{(1)} \]
\[ \pi r^2 h = 924 \quad \text{(2)} \]

3. **Divide equation (1) by equation (2)**:
\[ \frac{2\pi rh}{\pi r^2 h} = \frac{264}{924} \]
- Simplifying the left side:
\[ \frac{2}{r} = \frac{264}{924} \]

4. **Calculate the right side**:
- Simplifying \(\frac{264}{924}\):
\[ \frac{264 \div 132}{924 \div 132} = \frac{2}{7} \]
- Therefore, we have:
\[ \frac{2}{r} = \frac{2}{7} \]

5. **Cross-multiply to find \(r\)**:
\[ 2 \cdot 7 = 2 \cdot r \implies r = 7 \text{ m} \]

6. **Substitute \(r\) back to find \(h\)**:
- Using equation (1):
\[ 2\pi(7)h = 264 \]
- Simplifying:
\[ 14\pi h = 264 \]
- Dividing both sides by \(14\pi\):
\[ h = \frac{264}{14\pi} \]
- Using \(\pi \approx \frac{22}{7}\):
\[ h = \frac{264}{14 \times \frac{22}{7}} = \frac{264 \times 7}{14 \times 22} \]
- Simplifying:
\[ h = \frac{1848}{308} = 6 \text{ m} \]

7. **Calculate the diameter**:
- The diameter \(d\) is given by:
\[ d = 2r = 2 \times 7 = 14 \text{ m} \]

8. **Find the ratio of height to diameter**:
- The ratio of height \(h\) to diameter \(d\) is:
\[ \text{Ratio} = \frac{h}{d} = \frac{6}{14} = \frac{3}{7} \]

### Final Answer:
The ratio of the height to the diameter of the cylinder is \( \frac{3}{7} \).