The problem you're dealing with involves a cube whose edges are expanding at a rate of 6 centimeters per second. We are asked to determine how fast the volume of the cube is changing when each edge is 2 centimeters and 10 centimeters.

Step-by-Step Solution:

1. Volume of a Cube Formula:
   The volume $V$ of a cube is given by: $V = s^3$ where $s$ is the length of an edge of the cube.

2. Given Rate of Change:
   The rate at which the edge length changes is given as: $\frac{ds}{dt} = 6 \text{ cm/sec}$ where $s$ is the length of the edge and $\frac{ds}{dt}$ is the rate of change of the edge length.

3. Differentiate the Volume Formula:
   To find how fast the volume is changing, we differentiate the volume equation with respect to time $t$: $\frac{dV}{dt} = 3s^2 \cdot \frac{ds}{dt}$ where $\frac{dV}{dt}$ is the rate of change of the volume.

4. Substitute Known Values:
   Now, we can substitute the given values into the differentiated equation.

   • (a) When $s = 2$ cm: $\frac{dV}{dt} = 3(2)^2 \cdot 6 = 3 \cdot 4 \cdot 6 = 72 \text{ cubic centimeters per second.}$

   • (b) When $s = 10$ cm: $\frac{dV}{dt} = 3(10)^2 \cdot 6 = 3 \cdot 100 \cdot 6 = 1800 \text{ cubic centimeters per second.}$

Final Answers:

• (a) When the edge is 2 cm, the volume is changing at 72 cubic centimeters per second.
• (b) When the edge is 10 cm, the volume is changing at 1800 cubic centimeters per second.

Would you like any further clarifications or additional questions related to this? Here's a list of related questions you might find interesting:

1. How does the rate of change of volume vary with different edge lengths?
2. Can you calculate the rate of change in surface area given the same edge length expansion rate?
3. What is the physical significance of the differentiation in this type of problem?
4. How does the volume of a cube relate to its surface area mathematically?
5. What happens to the rate of change if the edge length rate of change is halved?

Tip: When dealing with related rates, always express the variables in terms of each other, and then differentiate with respect to time, keeping track of the rates you're given.