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

Options

A. `V^3 = 216 S^2`
B. `S^3 = 216 V^2`
C. `S^3 = 6 V^2`
D. `S^2 = 36 V^3`

Answer

To solve the problem, we need to find the relationship between the total surface area \( S \) and the volume \( V \) of a cube. Let's break this down step by step.

### Step 1: Understand the definitions
- The **total surface area** \( S \) of a cube with side length \( A \) is given by the formula:
\[ S = 6A^2 \]
- The **volume** \( V \) of a cube with side length \( A \) is given by the formula:
\[ V = A^3 \]

### Step 2: Express \( A \) in terms of \( S \)
From the formula for the total surface area, we can express \( A^2 \) in terms of \( S \):
\[ A^2 = \frac{S}{6} \]

### Step 3: Express \( A \) in terms of \( V \)
From the formula for the volume, we can express \( A \) in terms of \( V \):
\[ A = V^{1/3} \]

### Step 4: Substitute \( A \) into the surface area equation
Now, we can substitute \( A \) from the volume equation into the surface area equation. First, we need \( A^2 \):
\[ A^2 = (V^{1/3})^2 = V^{2/3} \]

### Step 5: Set the equations equal
Now we can set the two expressions for \( A^2 \) equal to each other:
\[ \frac{S}{6} = V^{2/3} \]

### Step 6: Rearranging the equation
Multiplying both sides by 6 gives us:
\[ S = 6V^{2/3} \]

### Step 7: Cubing both sides
To find a relationship involving \( S^3 \) and \( V^2 \), we cube both sides:
\[ S^3 = (6V^{2/3})^3 \]
\[ S^3 = 216V^2 \]

### Conclusion
Thus, the relationship we derived is:
\[ S^3 = 216V^2 \]
This means the correct option is:
- **Option 2:** \( S^3 = 216V^2 \)