https://file.aiquickdraw.com/mathbot/file/17301778405055r83zybm.jpg

To solve this problem, let’s go through each part step-by-step.

### Part (a)
The problem specifies that squares of side \( x \) are cut from each corner of a \( 24 \, \text{in} \) by \( 8 \, \text{in} \) metal sheet to form an open box.

1. **Length of the box after cutting:**  
   Original length of the sheet is \( 24 \, \text{in} \), and since \( x \)-inch squares are cut from both ends, the new length becomes:
   \[
   \text{Length} = 24 - 2x
   \]

2. **Width of the box after cutting:**  
   Original width of the sheet is \( 8 \, \text{in} \), and similarly, cutting \( x \)-inch squares from both ends reduces the width to:
   \[
   \text{Width} = 8 - 2x
   \]

3. **Height of the box:**  
   The height of the box is determined by the side length \( x \) of the squares cut out from each corner. So, the height is:
   \[
   \text{Height} = x
   \]

### Part (b)
Now, we need to find the volume \( V \) of the box in terms of \( x \).

The volume of a rectangular box is given by:
\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]

Substituting the expressions we found in part (a):
\[
V = (24 - 2x)(8 - 2x)(x)
\]

So, the volume function for the box is:
\[
V = x(24 - 2x)(8 - 2x)
\]

Let me know if you would like further details on simplifying this expression, or have any questions!

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Here are 5 follow-up questions for deeper understanding:

1. What would happen to the volume of the box if \( x \) becomes too large?
2. How does changing \( x \) affect the box's dimensions?
3. What is the maximum possible value of \( x \) for this box to remain physically feasible?
4. Can you find the derivative of \( V \) to determine the value of \( x \) that maximizes the volume?
5. How would the volume function change if the original metal sheet dimensions were different?

**Tip:** Remember that for open-box problems, cutting too large of a square from each corner may result in a negative dimension. Always verify feasible values for \( x \).

A metalworker wants to make an open box from a sheet of metal, by cutting equal squares from each corner. Write expressions for the length, width, and height of the open box and create a volume function.

Learn how to calculate the volume of an open box made by cutting equal squares from each corner of a metal sheet. This involves using algebraic expressions for length, width, and height and applying the volume formula. Suitable for high school students in Grades 9-11.

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