The volume of a sphere is given by the formula `4 V=pir^3` , where r is the radius of the sphere.Which of the following gives the radius of the sphere in terms of the volume of the sphere?

Options

A. `(4pi)/(3V)`
B. `(3V)/(4pi)`
C. `root3((4pi)/(3V))`
D. `root3((3V)/(4pi))`

Explanation

To find the radius of a sphere in terms of its volume, we start with the formula for the volume of a sphere, which is given by:

\[ V = \frac{4}{3} \pi r^3 \]

### Step 1: Rearranging the Volume Formula
We want to isolate \( r \) in the equation. To do this, we first multiply both sides of the equation by \( \frac{3}{4} \) to eliminate the fraction:

\[ \frac{3}{4} V = \frac{3}{4} \left( \frac{4}{3} \pi r^3 \right) \]

### Step 2: Simplifying the Equation
On the right side, the \( \frac{3}{4} \) and \( \frac{4}{3} \) will cancel each other out:

\[ \frac{3}{4} V = \pi r^3 \]

### Step 3: Dividing by \( \pi \)
Next, we divide both sides by \( \pi \) to isolate \( r^3 \):

\[ \frac{3V}{4\pi} = r^3 \]

### Step 4: Taking the Cube Root
To solve for \( r \), we take the cube root of both sides:

\[ r = \sqrt[3]{\frac{3V}{4\pi}} \]

Thus, the radius \( r \) of the sphere in terms of its volume \( V \) is:

\[ r = \sqrt[3]{\frac{3V}{4\pi}} \]

### Conclusion
The final expression gives the radius of the sphere in terms of its volume.

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