Hello, Magentarita!

A closed box with a square base is required to have a volume of 10 ft³

(a) Express the amount $\displaystyle A$ of material used to make such a box

as a function of the length $\displaystyle x$ of a side of the square base. Code:

*-----*
/ /|
/ / | y
*-----* |
| | *
y | | /
| |/ x
*-----*
x

The volume is 10 ft³: .$\displaystyle V \:=\:x^2y \:=\:10 \quad\Rightarrow\quad y \:=\:\frac{10}{x^2}$ .[1]

The surface area is: .$\displaystyle A \;=\;2x^2 + 4xy$ .[2]

Substitute [1] into [2]: .$\displaystyle A \;=\;2x^2 + 4x\left(\frac{10}{x^2}\right)$

Therefore: .$\displaystyle \boxed{A \;=\;2x^2 + \frac{40}{x}}$

(b) How much material is required for a base 1 ft by 1 ft? If $\displaystyle x = 1\!:\;\;A \;=\;2(1^2) + \frac{40}{1} \;=\;\boxed{42\text{ ft}^2}$