A particular sphere has the property that its surface area has the samme numerical value as its volume. What is the length of the radius of this sphere?

Options

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

Explanation

To find the radius of the sphere where its surface area is numerically equal to its 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 the formula:
\[ A = 4\pi r^2 \]
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]

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

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

### Step 4: Rearrange the equation.
To isolate \( r \), we can rearrange the equation:
\[ 3r^2 = r^3 \]

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

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

Thus, the radius of the sphere is:
\[ \boxed{3} \]

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