The magnitude of surface area of sphere is half the magnitude of the volume of the sphere . Find the diameter of the sphere .

Options

A. 3 units
B. 6 units
C. 12 units
D. 18 units

Answer

To find the diameter of the sphere given that the surface area is half the volume, we can follow these steps:

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

### Step 2: Set up the equation based on the problem statement.
According to the problem, the surface area is half of the volume:
\[ 4\pi r^2 = \frac{1}{2} \left( \frac{4}{3}\pi r^3 \right) \]

### Step 3: Simplify the equation.
First, simplify the right side:
\[ 4\pi r^2 = \frac{2}{3}\pi r^3 \]
Now, we can eliminate \( \pi \) from both sides:
\[ 4r^2 = \frac{2}{3} r^3 \]

### Step 4: Rearrange the equation.
Multiply both sides by 3 to eliminate the fraction:
\[ 12r^2 = 2r^3 \]
Now, divide both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[ 12 = 2r \]

### Step 5: Solve for the radius \( r \).
Now, divide both sides by 2:
\[ r = 6 \]

### Step 6: Find the diameter of the sphere.
The diameter \( d \) is twice the radius:
\[ d = 2r = 2 \times 6 = 12 \]

### Final Answer:
The diameter of the sphere is \( 12 \) units.

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