The curved surface area of a cylinder pillar is `264 m^(2)` and its volume is `924 m^(3)`.
(Taking `pi = (22)/(7)`). Find the ratio of its diameter to its height.

Options

A. `14:6`
B. `6:7`
C. `3:7`
D. `7:3`

Answer

To solve the problem, we need to find the ratio of the diameter to the height of a cylindrical pillar given its curved surface area and volume.

### Step-by-Step Solution:

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

2. **Substitute the values**:
- We know that the CSA is \(264 \, m^2\) and the volume is \(924 \, m^3\).
- Therefore, we have:
\[ 2\pi rh = 264 \]
\[ \pi r^2 h = 924 \]

3. **Divide the two equations**:
- Divide the CSA equation by the volume equation:
\[ \frac{2\pi rh}{\pi r^2 h} = \frac{264}{924} \]
- Simplifying the left side, we get:
\[ \frac{2}{r} = \frac{264}{924} \]

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

5. **Solve for r**:
- Cross-multiply to find \(r\):
\[ 2 \cdot 7 = 2 \cdot r \implies r = 7 \, m \]

6. **Find the height (h)**:
- Substitute \(r = 7\) back into the CSA formula:
\[ 2\pi rh = 264 \]
- Plugging in the values:
\[ 2 \cdot \frac{22}{7} \cdot 7 \cdot h = 264 \]
- Simplifying:
\[ 44h = 264 \]
- Solving for \(h\):
\[ h = \frac{264}{44} = 6 \, m \]

7. **Calculate the diameter**:
- The diameter \(d\) is given by:
\[ d = 2r = 2 \cdot 7 = 14 \, m \]

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

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