The problem provides the formula for the volume of a sphere:

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

We are given that the volume of the bouncy ball is $4200\pi \, \text{mm}^3$, and we need to find the radius $r$, keeping the answer to 2 decimal places if necessary.

Step-by-Step Solution:

1. The volume formula is:

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

2. Substitute the given volume $V = 4200 \pi \, \text{mm}^3$ into the formula:

$4200 \pi = \frac{4}{3} \pi r^3$

3. Divide both sides by $\pi$ to simplify:

$4200 = \frac{4}{3} r^3$

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

$r^3 = 4200 \times \frac{3}{4} = 3150$

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

$r = \sqrt[3]{3150}$

6. Using a calculator to find the cube root:

$r \approx 14.72 \, \text{mm}$

Thus, the radius of the bouncy ball is approximately $\boxed{14.72 \, \text{mm}}$.

Would you like more details on any step, or do you have any other questions?

5 Related Questions:

1. How is the volume of a cylinder different from that of a sphere?
2. What is the formula for the surface area of a sphere, and how would you apply it here?
3. How would the result change if the volume was given in $\text{cm}^3$ instead of $\text{mm}^3$?
4. How would you solve this problem if the sphere was hollow with a certain thickness?
5. What is the relationship between the radius and diameter of a sphere?

Tip:

When dealing with sphere-related problems, always ensure your units are consistent throughout your calculations.