The volume and the surface area of a sphere are numerically equal, then the radius of sphere is

Options

A. 0 units
B. 1 units
C. 2 units
D. 3 units

Answer

To solve the problem where the volume and surface area of a sphere are numerically equal, we will follow these steps:

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

### Step 2: Set the volume equal to the surface area.
According to the problem, the volume and surface area are numerically equal:
\[ \frac{4}{3} \pi r^3 = 4 \pi r^2 \]

### Step 3: Simplify the equation.
We can simplify this equation by dividing both sides by \( 4 \pi \) (assuming \( r \neq 0 \)):
\[ \frac{1}{3} r^3 = r^2 \]

### Step 4: Rearrange the equation.
To eliminate \( r^2 \) from the right side, we can multiply both sides by 3:
\[ r^3 = 3r^2 \]

### Step 5: Factor the equation.
Now, we can factor out \( r^2 \):
\[ r^2 (r - 3) = 0 \]

### Step 6: Solve for \( r \).
Setting each factor to zero gives us:
1. \( r^2 = 0 \) (which implies \( r = 0 \), but this is not a valid solution for a sphere)
2. \( r - 3 = 0 \) (which gives \( r = 3 \))

### Conclusion:
Thus, the radius of the sphere is:
\[ r = 3 \text{ units} \]

---