To solve this problem, we need to go through each part step-by-step.

Part (a): Expressions for Length, Width, and Height

1. Length: The original length of the metal sheet is 24 inches. When squares of side length $x$ are cut out from each corner, the length of the box becomes: $\text{Length} = 24 - 2x$

2. Width: The original width of the metal sheet is 18 inches. After cutting out squares from each corner, the width of the box is: $\text{Width} = 18 - 2x$

3. Height: The height of the box will be equal to the side length of the squares cut out, so: $\text{Height} = x$

Part (b): Volume Function

The volume $V$ of the box is given by the formula: $V = \text{Length} \times \text{Width} \times \text{Height}$ Substituting the expressions we found: $V = (24 - 2x)(18 - 2x)(x)$ Expanding and simplifying this function gives us the volume function in terms of $x$.

Part (c): Graphing the Function and Finding Maximum Volume

To find the maximum volume, graph $V(x) = (24 - 2x)(18 - 2x)x$ and determine the value of $x$ that gives the highest point on the graph within a reasonable domain. Typically, this maximum occurs within a restricted range of $x$ (where $x$ is less than half of the smaller dimension of the metal sheet).

The answer involves choosing the correct graph, which likely corresponds to the graph that shows a maximum point for volume $V$ within the specified window. From the given options, we will need to analyze which graph aligns with our calculated maximum.

Now let me know if you’d like more detailed calculations for the volume function or assistance with specific parts!

Here are some related questions to deepen understanding:

1. What is the maximum possible value of $x$ for this setup?
2. How does changing the initial dimensions of the metal sheet affect the volume?
3. What happens to the volume if the height $x$ is increased slightly beyond the optimal point?
4. Can we find the maximum volume using calculus, and how would that approach look?
5. What are practical applications of this kind of problem in manufacturing?

Tip: Remember, when optimizing volume for boxes, always check that your solution is within physical constraints (dimensions should remain positive).