The formula for the volume of a sphere is given by:

$V = \frac{4}{3} \pi r^3$

To solve for the radius $r$ in terms of the volume $V$, we can rearrange this formula as follows:

1. Multiply both sides by $\frac{3}{4\pi}$ to isolate $r^3$:

   $r^3 = \frac{3V}{4\pi}$

2. Now, take the cube root of both sides to solve for $r$:

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

Now, we can compare this result to the answer choices:

• Option B matches $\sqrt[3]{\frac{3V}{4\pi}}$, which is the correct form.

Thus, the answer is:

B. $\sqrt[3]{\frac{3V}{4\pi}}$

Would you like further details on this solution, or do you have any other questions?

Additional Questions for Deeper Understanding:

1. How would the formula change if we were dealing with the surface area of a sphere instead of volume?
2. Can you derive the formula for the surface area of a sphere using the radius found here?
3. What if the problem gave the diameter instead of the radius — how would that affect the volume formula?
4. How can this concept be applied in real-world scenarios involving spheres, like in calculating fuel tank volumes?
5. What happens to the volume of the sphere if the radius is doubled?

Tip:

When solving for one variable in terms of others, always isolate the desired variable first before applying inverse operations (like taking roots or logarithms) to simplify the expression effectively.