If the volume and surface area of a sphere are numerically the same, then its radius is

Options

A. 1 units
B. 2 units
C. 3 units
D. 4 units

Answer

To find the radius of a sphere when its volume and surface area are numerically the same, we can follow these steps:

### Step-by-Step Solution:

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:
\[ A = 4 \pi r^2 \]

2. **Set the volume equal to the surface area**:
Since we are given that the volume and surface area are numerically the same, we can set the two formulas equal to each other:
\[ \frac{4}{3} \pi r^3 = 4 \pi r^2 \]

3. **Cancel common terms**:
We can simplify the equation by canceling \( 4 \pi \) from both sides (as long as \( r \neq 0 \)):
\[ \frac{1}{3} r^3 = r^2 \]

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

5. **Factor the equation**:
We can factor out \( r^2 \) from the left side:
\[ r^2 (r - 3) = 0 \]

6. **Solve for \( r \)**:
This equation gives us two solutions:
- \( r^2 = 0 \) which implies \( r = 0 \) (not a valid solution for radius)
- \( r - 3 = 0 \) which implies \( r = 3 \)

Thus, the radius of the sphere is \( r = 3 \).

### Final Answer:
The radius of the sphere is \( 3 \) units.

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