2. Squares of side a are cut from each corner of a 8 in x 6 in rectangle, so that its sides can be folded to make a box with no top. Represent a function in terms of a that can define the volume of the box

To represent the volume of the box as a function of \( a \), let's go through the process step by step:

1. **Original Rectangle Dimensions:**
   The rectangle has dimensions 8 inches by 6 inches.

2. **Cutting Squares:**
   Squares of side \( a \) are cut from each corner. This means the length and width of the resulting base of the box will be reduced by twice the side of the squares (since a square is cut from both ends of each dimension).

   - New length = \( 8 - 2a \)
   - New width = \( 6 - 2a \)

3. **Folding to Form a Box:**
   When the sides are folded up to form the sides of the box, the height of the box will be \( a \).

4. **Volume of the Box:**
   The volume \( V \) of the box can be found by multiplying the length, width, and height.

Therefore, the volume \( V \) of the box as a function of \( a \) is:
\[
V(a) = (8 - 2a)(6 - 2a)a
\]

We can expand this function to get a clearer form:

\[
V(a) = a (8 - 2a)(6 - 2a)
\]

First, expand \( (8 - 2a)(6 - 2a) \):
\[
(8 - 2a)(6 - 2a) = 8 \cdot 6 - 8 \cdot 2a - 2a \cdot 6 + (2a)^2
\]
\[
= 48 - 16a - 12a + 4a^2
\]
\[
= 48 - 28a + 4a^2
\]

Now, multiply this by \( a \):

\[
V(a) = a(48 - 28a + 4a^2)
\]
\[
= 48a - 28a^2 + 4a^3
\]

So, the volume function in terms of \( a \) is:
\[
V(a) = 4a^3 - 28a^2 + 48a
\]

This function represents the volume of the box after squares of side \( a \) are cut from each corner and the sides are folded up to form the box.

---

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 domain of the function \( V(a) \)?
2. How do you find the maximum volume of the box?
3. Can you explain the significance of the cubic term \( 4a^3 \) in the volume function?
4. What happens to the volume as \( a \) approaches 0?
5. How does the volume change as \( a \) approaches its maximum possible value?
6. How can you verify if the volume function is correct?
7. What are the physical constraints that \( a \) must satisfy in this problem?
8. Can you graph the volume function \( V(a) \) and interpret its key features?

**Tip:** To find the maximum volume, you can use calculus by finding the critical points of \( V(a) \) through its first derivative and applying the second derivative test to determine concavity.

Learn how to calculate the volume of a box formed by cutting squares from the corners of an 8 in x 6 in rectangle. Detailed step-by-step solution provided, including the volume function in terms of side length 'a'. Suitable for high school students studying geometry and algebra.

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