The volume of a cube with surface area 384 square meter is :

Question

Accepted Answer

To determine the volume of a cube given its surface area, we need to follow these steps:

1. **Understanding the Surface Area of a Cube:**
   - A cube has 6 faces, each of which is a square with side length $ s $.
   - The surface area $ A $ of a cube is given by the formula: 
     $$
     A = 6s^2
     $$
   - We are given that the surface area is 384 square meters. Therefore, we have:
     $$
     6s^2 = 384
     $$

2. **Solving for the Side Length $ s $:**
   - Divide both sides of the equation by 6 to solve for $ s^2 $:
     $$
     s^2 = \frac{384}{6} = 64
     $$
   - Take the square root of both sides to find $ s $:
     $$
     s = \sqrt{64} = 8
     $$

3. **Calculating the Volume of the Cube:**
   - The volume $ V $ of a cube is given by the formula:
     $$
     V = s^3
     $$
   - Substitute $ s = 8 $ into the formula:
     $$
     V = 8^3 = 512
     $$

4. **Conclusion:**
   - The volume of the cube is $ 512~m^3 $, which corresponds to Option 4.

**Explanation of Why Other Options are Incorrect:**

- **Option 1: $216~m^{3}$**
  - If the volume were 216, then $ s^3 = 216 $. Solving for $ s $, we find $ s = \sqrt[3]{216} = 6 $. This would imply a surface area of $ 6 \times 6^2 = 216 $, which does not match the given surface area of 384.

- **Option 2: $256~m^{3}$**
  - If the volume were 256, then $ s^3 = 256 $. Solving for $ s $, we find $ s = \sqrt[3]{256} = 6.3496 $ (approximately). This would imply a surface area of $ 6 \times (6.3496)^2 \approx 242 $, which does not match the given surface area of 384.

- **Option 3: $484~m^{3}$**
  - If the volume were 484, then $ s^3 = 484 $. Solving for $ s $, we find $ s = \sqrt[3]{484} \approx 7.8 $. This would imply a surface area of $ 6 \times (7.8)^2 \approx 365 $, which does not match the given surface area of 384.

Therefore, based on the calculations and logical reasoning, Option 4 is indeed the correct answer, as it aligns perfectly with the given surface area and the derived side length of the cube.