Answer
The amount of material needed to construct the box in terms of the length of a side of its square base $x$ is, $A={{x}^{2}}+\frac{40}{x}$ square feet.
Work Step by Step
Volume of the provided open box $=10\text{ cubic feet}$
Now, use the formula of the volume of cuboid, $V=l\cdot w\cdot h$ in the above equation
$l\cdot w\cdot h=10$
Substitute $x$ for $l$ and $w$ , and $y$ for $h$ in the above equation
$\begin{align}
& x\cdot x\cdot y=10 \\
& {{x}^{2}}\cdot y=10
\end{align}$
Or
$y=\frac{10}{{{x}^{2}}}$
The surface area of given open box $A=lw+2lh+2wh$
Substitute $x$ for $l$ and $w$ , and $y$ for $h$ in the above equation
$\begin{align}
& A=x\cdot x+2\cdot x\cdot y+2\cdot x\cdot y \\
& ={{x}^{2}}+4\cdot x\cdot y
\end{align}$
Substitute $\frac{10}{{{x}^{2}}}$ for $y$ in the above equation
$A={{x}^{2}}+4\cdot x\cdot \frac{10}{{{x}^{2}}}$
Simplify
$A={{x}^{2}}+\frac{40}{x}$
The required amount of material needed to construct the box in terms of the length of a side of its square base $x$ is, $A={{x}^{2}}+\frac{40}{x}$ square feet.
