### Solution:
#### (a) Write a function [tex]\( V(x) \)[/tex] that represents the volume of the box.
The function representing the volume [tex]\( V \)[/tex] of the box is given by:
[tex]\[
V(x) = 4x^3 - 144x^2 + 1296x
\][/tex]
#### (b) Determine the domain of the function [tex]\( V \)[/tex].
To determine the domain of the function [tex]\( V \)[/tex], we need to consider the real-world context of the problem. Since it represents the volume of a box, [tex]\( x \)[/tex] (the height of the box) must be a non-negative value. Additionally, the volume must also be a non-negative value. Thus, we need to find [tex]\( x \)[/tex] such that [tex]\( V(x) \geq 0 \)[/tex].
Upon examining the function [tex]\( V(x) \)[/tex]:
[tex]\[
V(x) = 4x^3 - 144x^2 + 1296x
\][/tex]
Since this is a cubic polynomial, it is defined for all real numbers. However, keeping the practical constraints in mind (i.e., [tex]\( x \)[/tex] must be positive and the volume cannot be negative), the domain is all positive real numbers:
[tex]\[
\text{Domain: } (0, \infty)
\][/tex]
#### (c) Use a graphing utility to construct a table that shows the box heights [tex]\( x \)[/tex] and the corresponding volumes [tex]\( V(x) \)[/tex].
We will construct a table that displays the height [tex]\( x \)[/tex] and the corresponding volume [tex]\( V(x) \)[/tex] for specific values of [tex]\( x \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x \text{ (in)} & V(x) \text{ (in}^3\text{)} \\
\hline
1 & 1156 \\
\hline
2 & 2048 \\
\hline
3 & 2700 \\
\hline
4 & 3136 \\
\hline
5 & 3380 \\
\hline
6 & 3456 \\
\hline
\end{array}
\][/tex]
So, the table is filled with the values calculated for each height [tex]\( x \)[/tex]:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$, in & $V(x)$, in $^3$ \\
\hline
1 & 1156 \\
\hline
2 & 2048 \\
\hline
3 & 2700 \\
\hline
4 & 3136 \\
\hline
5 & 3380 \\
\hline
6 & 3456 \\
\hline
\end{tabular}
\][/tex]
