If the area of one face of the cube is 1.5 times of the perimeter (in value) of the same face, then volume of the cube is :

Question

Accepted Answer

To determine why Option 3 is the correct answer, we need to analyze the problem using the given information about the cube.

### Explanation:

1. **Understanding the Problem:**
   - We are given that the area of one face of the cube is 1.5 times the perimeter of the same face.
   - Let the side length of the cube be $ s $.

2. **Calculating the Area and Perimeter:**
   - The area of one face of the cube is $ s^2 $.
   - The perimeter of one face of the cube is $ 4s $.

3. **Setting Up the Equation:**
   - According to the problem, the area is 1.5 times the perimeter:
     $$
     s^2 = 1.5 \times 4s
     $$
   - Simplifying the equation:
     $$
     s^2 = 6s
     $$
   - Dividing both sides by $ s $ (assuming $ s \neq 0 $):
     $$
     s = 6
     $$

4. **Calculating the Volume:**
   - The volume of the cube is given by $ s^3 $.
   - Substituting $ s = 6 $:
     $$
     \text{Volume} = 6^3 = 216
     $$

5. **Conclusion:**
   - Therefore, the volume of the cube is 216 units, which corresponds to Option 3.

### Why Other Options are Incorrect:

- **Option 1 (125 units):**
  - If the side length $ s $ were such that the volume is 125, then $ s^3 = 125 $ implies $ s = 5 $. However, substituting $ s = 5 $ into the equation $ s^2 = 6s $ does not hold true, as $ 25 \neq 30 $.

- **Option 2 (144 units):**
  - If the side length $ s $ were such that the volume is 144, then $ s^3 = 144 $ does not yield an integer value for $ s $, and thus cannot satisfy the equation $ s^2 = 6s $.

- **Option 4 (312 units):**
  - If the side length $ s $ were such that the volume is 312, then $ s^3 = 312 $ does not yield an integer value for $ s $, and thus cannot satisfy the equation $ s^2 = 6s $.

By following the logical steps and calculations, we confirm that Option 3 is indeed the correct answer, as it satisfies all the conditions given in the problem.