a) Letting $x$ denote the length of one of the sides of the bottom of the box, and letting $y$ denote the height of the box, sketch a picture of the box, labeling the length, width, and height. The company wants to use exactly 121 cm$^{2}$ of material for the box. That is, the box's surface area will be equal to 121 cm$^{2}$. Write an equation in terms of $x$ and $y$ representing this constraint, putting the variables on the left side.
$=$ cm$^{2}$

Solve for $y$ to find $y$ as a function of $x$.
$y=$

b) Using your work from part (a), write a formula for the volume of the box, $V(x)$, as a function of $x$. Simplify as much as possible.
$V(x)=$

c) Use an online source to approximate the maximum value of $V(x)$ on the interval $[0,10]$. What $x$ value corresponds to this maximum value?

What does this mean in the context of the problem?

A. This is the maximum volume of a box with a surface area of 121 cm$^2$.
B. This is the length of the side of the box that produces the maximum volume of a box with a surface area of 121 cm$^2$.
C. This is the area of the base of the box that produces the maximum volume of a box with a surface area of 121 cm$^2$.

Hint: