If surface area and volume of a sphere are S and V respectively, then value of `(S^(3))/(V^(2))` is

Options

A. `36 pi` units
B. `9 pi` units
C. `18 pi` units
D. `27 pi` units

Answer

To find the value of \(\frac{S^3}{V^2}\) where \(S\) is the surface area and \(V\) is the volume of a sphere, we can follow these steps:

### Step 1: Write the formulas for surface area and volume of a sphere
The surface area \(S\) of a sphere is given by the formula:
\[ S = 4\pi r^2 \]
The volume \(V\) of a sphere is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]

### Step 2: Substitute \(S\) and \(V\) into the expression \(\frac{S^3}{V^2}\)
We need to calculate:
\[ \frac{S^3}{V^2} = \frac{(4\pi r^2)^3}{\left(\frac{4}{3}\pi r^3\right)^2} \]

### Step 3: Calculate \(S^3\)
Calculating \(S^3\):
\[ S^3 = (4\pi r^2)^3 = 4^3 \cdot \pi^3 \cdot (r^2)^3 = 64\pi^3 r^6 \]

### Step 4: Calculate \(V^2\)
Calculating \(V^2\):
\[ V^2 = \left(\frac{4}{3}\pi r^3\right)^2 = \left(\frac{4}{3}\right)^2 \cdot \pi^2 \cdot (r^3)^2 = \frac{16}{9}\pi^2 r^6 \]

### Step 5: Substitute \(S^3\) and \(V^2\) into the expression
Now substituting \(S^3\) and \(V^2\) into the expression:
\[ \frac{S^3}{V^2} = \frac{64\pi^3 r^6}{\frac{16}{9}\pi^2 r^6} \]

### Step 6: Simplify the expression
The \(r^6\) terms cancel out:
\[ \frac{S^3}{V^2} = \frac{64\pi^3}{\frac{16}{9}\pi^2} = 64\pi^3 \cdot \frac{9}{16\pi^2} \]
This simplifies to:
\[ = \frac{64 \cdot 9}{16} \cdot \pi = 36\pi \]

### Final Answer
Thus, the value of \(\frac{S^3}{V^2}\) is:
\[ \boxed{36\pi} \]