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

Options

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

Explanation

To find the diameter of a sphere when its volume is numerically equal to its surface area, we can 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.
Since the problem states that the volume is numerically equal to the surface area, we can set the two equations equal to each other:
\[ \frac{4}{3} \pi r^3 = 4 \pi r^2 \]

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

### Step 4: Rearrange the equation.
To simplify further, we can multiply both sides by 3 to eliminate the fraction:
\[ r^3 = 3r^2 \]
Now, we can rearrange this to:
\[ r^3 - 3r^2 = 0 \]

### Step 5: Factor the equation.
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 \( r = 0 \), not a valid solution for a sphere)
2. \( r - 3 = 0 \) (which implies \( r = 3 \))

### Step 7: Calculate the diameter.
The diameter \( d \) of the sphere is given by:
\[ d = 2r \]
Substituting \( r = 3 \):
\[ d = 2 \times 3 = 6 \text{ cm} \]

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

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