If S is the total surface area of a cube and V is its volume, then which one of the following is correct?

Question

Accepted Answer

To determine why Option 2 is correct, let's analyze the relationship between the total surface area $ S $ and the volume $ V $ of a cube.

For a cube with side length $ a $:
- The total surface area $ S $ is given by $ S = 6a^2 $.
- The volume $ V $ is given by $ V = a^3 $.

Now, let's explore Option 2: $ S^3 = 216 V^2 $.

### Verification of Option 2:
1. **Express $ S $ and $ V $ in terms of $ a $:**
   - $ S = 6a^2 $
   - $ V = a^3 $

2. **Calculate $ S^3 $:**
   $$
   S^3 = (6a^2)^3 = 216a^6
   $$

3. **Calculate $ 216 V^2 $:**
   $$
   216 V^2 = 216(a^3)^2 = 216a^6
   $$

Since $ S^3 = 216a^6 $ and $ 216 V^2 = 216a^6 $, we have $ S^3 = 216 V^2 $. Therefore, Option 2 is correct.

### Explanation of Why Other Options are Incorrect:

1. **Option 1: $ V^3 = 216 S^2 $**
   - Calculate $ V^3 $:
     $$
     V^3 = (a^3)^3 = a^9
     $$
   - Calculate $ 216 S^2 $:
     $$
     216 S^2 = 216(6a^2)^2 = 216 \times 36a^4 = 7776a^4
     $$
   - Clearly, $ a^9 \neq 7776a^4 $, so Option 1 is incorrect.

2. **Option 3: $ S^3 = 6 V^2 $**
   - We already calculated $ S^3 = 216a^6 $.
   - Calculate $ 6 V^2 $:
     $$
     6 V^2 = 6(a^3)^2 = 6a^6
     $$
   - Since $ 216a^6 \neq 6a^6 $, Option 3 is incorrect.

3. **Option 4: $ S^2 = 36 V^3 $**
   - Calculate $ S^2 $:
     $$
     S^2 = (6a^2)^2 = 36a^4
     $$
   - Calculate $ 36 V^3 $:
     $$
     36 V^3 = 36(a^3)^3 = 36a^9
     $$
   - Since $ 36a^4 \neq 36a^9 $, Option 4 is incorrect.

In conclusion, the only option that correctly represents the relationship between the total surface area and the volume of a cube is Option 2: $ S^3 = 216 V^2 $.