The curved surface of a cylindrical pillar is $264 m^2$ and its volume is $924 m^3$. The diameter is equal to :

Question

Accepted Answer

To determine why Option 2 (14 m) is the correct answer for the diameter of the cylindrical pillar, we need to use the formulas for the curved surface area and volume of a cylinder.

### Given:
- Curved Surface Area (CSA) = $264 \, m^2$
- Volume (V) = $924 \, m^3$

### Formulas:
1. Curved Surface Area of a Cylinder: $CSA = 2\pi rh$
2. Volume of a Cylinder: $V = \pi r^2 h$

Where:
- $r$ is the radius of the cylinder
- $h$ is the height of the cylinder
- $\pi$ is approximately 3.14159

### Step-by-Step Explanation:

1. **Using the Curved Surface Area:**
   $$
   2\pi rh = 264
   $$
   $$
   rh = \frac{264}{2\pi} = \frac{132}{\pi}
   $$

2. **Using the Volume:**
   $$
   \pi r^2 h = 924
   $$
   $$
   r^2 h = \frac{924}{\pi}
   $$

3. **Relating the Two Equations:**
   From the CSA equation, we have $h = \frac{132}{\pi r}$.
   Substitute $h$ in the volume equation:
   $$
   r^2 \left(\frac{132}{\pi r}\right) = \frac{924}{\pi}
   $$
   $$
   132r = 924
   $$
   $$
   r = \frac{924}{132} = 7
   $$

4. **Finding the Diameter:**
   The diameter $d$ is twice the radius:
   $$
   d = 2r = 2 \times 7 = 14 \, m
   $$

Thus, the diameter of the cylindrical pillar is 14 m, which corresponds to Option 2.

### Explanation of Incorrect Options:

- **Option 1 (7 m):** This is the radius, not the diameter. The diameter is twice the radius, so this option is incorrect.
- **Option 3 (21 m):** This value is larger than the calculated diameter. Using the given formulas and calculations, 21 m does not satisfy both the surface area and volume conditions.
- **Option 4 (3.5 m):** This value is half of the radius, which is incorrect for the diameter. It does not satisfy the conditions set by the problem.

Therefore, Option 2 (14 m) is the correct answer for the diameter of the cylindrical pillar.