To find a simple expression for the volume of a box made by cutting squares from the corners of an 8-inch by 6-inch rectangle and folding up the sides, we can follow these steps:

1. Initial Rectangle Dimensions:

   • Length: 8 inches
   • Width: 6 inches

2. Squares Cut from Each Corner:

   • Side of each square: $a$

3. New Dimensions of the Box's Base:

   • New length: $8 - 2a$
   • New width: $6 - 2a$

4. Height of the Box:

   • Height: $a$ (the same as the side of the square cut out)

5. Volume of the Box:

   • Volume $V$ = (length) $\times$ (width) $\times$ (height)
   • So, $V(a) = (8 - 2a)(6 - 2a)a$

This gives us the volume of the box as a function of $a$:

$V(a) = (8 - 2a)(6 - 2a)a$

To summarize:

$V(a) = (8 - 2a)(6 - 2a)a$

This is the function that defines the volume of the box in terms of $a$.

---

Would you like details or have any questions about this solution? Here are 8 related questions to further explore the topic:

1. What is the maximum possible value of $a$ for which the box can be formed?
2. How does the volume change as $a$ increases from 0 to its maximum value?
3. What is the volume of the box when $a = 1$ inch?
4. How can you find the value of $a$ that gives the maximum volume of the box?
5. What are the dimensions of the box when $a$ is at its maximum value for volume?
6. How do you determine if the volume function $V(a)$ is correct?
7. Can you graph the volume function to visualize its behavior?
8. What practical applications can this volume function have in real-world scenarios?

Tip: To solve for the maximum volume, you can use calculus techniques, such as finding the derivative of the volume function $V(a)$ and setting it to zero to find the critical points.