If the volume of a sphere is numerically equal to its surface area then its diameter is

Options

A. 6 cm
B. 4 cm
C. 2 cm
D. 3 cm

Answer

To solve the problem, we need to find the diameter of a sphere when its volume is numerically equal to its surface area.

### 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 the problem states that the volume is numerically equal to the surface area, we can write:
\[ \frac{4}{3} \pi r^3 = 4 \pi r^2 \]

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

4. **Rearrange the equation:**
Multiply both sides by 3 to eliminate the fraction:
\[ r^3 = 3r^2 \]
Now, we can rearrange this to:
\[ r^3 - 3r^2 = 0 \]

5. **Factor the equation:**
Factor out \( r^2 \):
\[ r^2 (r - 3) = 0 \]

6. **Solve for \( r \):**
Setting each factor to zero gives us:
- \( r^2 = 0 \) which implies \( r = 0 \) (not a valid solution for a sphere)
- \( r - 3 = 0 \) which implies \( r = 3 \)

7. **Find the diameter:**
The diameter \( d \) of a sphere is given by:
\[ d = 2r \]
Substituting \( r = 3 \):
\[ d = 2 \times 3 = 6 \]

### Final Answer:
The diameter of the sphere is \( 6 \) centimeters.

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