To determine how fast the volume of a sphere is increasing when the diameter is 60 mm, we need to use related rates in calculus.

First, we identify the variables and their relationships:

• Let $r$ be the radius of the sphere in mm.
• Let $V$ be the volume of the sphere in mm³.
• The relationship between the volume and the radius of a sphere is given by the formula: $V = \frac{4}{3} \pi r^3$
• Given: The radius $r$ is increasing at a rate of $\frac{dr}{dt} = 5 \text{ mm/s}$.

We need to find how fast the volume $V$ is increasing, i.e., $\frac{dV}{dt}$, when the diameter is 60 mm.

First, convert the diameter to the radius: $r = \frac{\text{diameter}}{2} = \frac{60 \text{ mm}}{2} = 30 \text{ mm}$

Next, we differentiate the volume formula with respect to time $t$: $V = \frac{4}{3} \pi r^3$ Taking the derivative with respect to $t$: $\frac{dV}{dt} = \frac{d}{dt} \left( \frac{4}{3} \pi r^3 \right) = 4 \pi r^2 \frac{dr}{dt}$

Now, we substitute $r = 30 \text{ mm}$ and $\frac{dr}{dt} = 5 \text{ mm/s}$ into the differentiated formula: $\frac{dV}{dt} = 4 \pi (30)^2 (5)$

Calculate the value: $\frac{dV}{dt} = 4 \pi (900) (5) = 18000 \pi$

Finally, round to two decimal places: $\frac{dV}{dt} \approx 18000 \times 3.14 = 56520 \text{ mm}^3/\text{s}$

Therefore, the volume of the sphere is increasing at approximately $56520 \text{ mm}^3/\text{s}$ when the diameter is 60 mm.

Would you like more details or have any questions?

Here are 8 relative questions that expand on this information:

1. How do you find the rate of change of the surface area of a sphere?
2. What is the relationship between the radius and the diameter of a sphere?
3. How does the rate of change of the radius affect the volume of a sphere?
4. How would the volume change if the radius was increasing at a different rate?
5. What are some practical applications of related rates in real-life scenarios?
6. How do you differentiate more complex volume formulas with respect to time?
7. What is the significance of using $\pi$ in the volume formula for a sphere?
8. How would you calculate the rate of change of the volume if the diameter was given instead of the radius?

Tip: When dealing with related rates problems, always identify the variables, their rates of change, and how they are related through differentiation.