Find the largest box that can be made from a fixed amount of material

Find the largest open box (i.e., no material covering the top face) that can be made from a rectangular piece of material by removing squares at each corner and turning up the sides in the following cases:

the rectangular material has edges 10 and 10;
the rectangular material has edges 12 and 18.

We begin with a rectangle with sides each of length 10. The picture is:

So, the edges of the base of the box are each of length $l = w = 10 - 2x$ , and the height is $x$ ; therefore, the volume is

$V = x(10-2x)^2 = 4x^3 - 40x^2 + 100x.$

Taking the derivative,

$\frac{dV}{dx} = 12x^2 - 80x + 100.$

Setting this equal to zero and solving,

$12x^2 - 80x + 100 = 0 \quad \implies \quad x = 5 \text{ or } \frac{5}{3}.$

Then looking at the sign of the derivative around these critical points we have

$\begin{align*} \frac{dV}{dx} &> 0 & \text{when} && 0 &< x < \frac{5}{3} \\ \frac{dV}{dx} &< 0 & \text{when} && \frac{5}{3} &< x < 5. \end{align*}$

Therefore, $V$ is increasing on $0 < x < \frac{5}{3}$ and decreasing on $\frac{5}{3} < x < 5$ . Hence, $V$ has a maximum at $x = \frac{5}{3}$ . Solving for the length and width of the base of the box we then have

$l = w = 10 - 2x = 10 - 2 \left( \frac{5}{3} \right) = \frac{20}{3}.$
This time we begin with a rectangle with length 18 and width 12. The picture is as follows:

So, the edges of the base of the box have lengths $w = 12-2x$ and $l = 18-2x$ , and the height of the box is $x$ . Thus, the volume is

$V = x (12-2x)(18-2x) = 4x^3 - 60x^2 + 216x.$

Taking the derivative,

$\frac{dV}{dx} = 12x^2 - 120x + 216.$

Setting this equal to 0 we obtain the critical points,

$12x^2 - 120x + 216 = 0 \quad \implies \quad x = 5 - \sqrt{7} \text{ or } 5 + \sqrt{7}.$

Then,

$\begin{align*} \frac{dV}{dx} &>0 & \text{when} && 0 &< x < 5 - \sqrt{7} \\ \frac{dV}{dx} &<0 & \text{when} && 5-\sqrt{7} & < x < 5 + \sqrt{7}. \end{align*}$

Hence, $V$ has a maximum at $x = 5 - \sqrt{7}$ . Solving for $l$ and $w$ we then have

$l = 18-2(5 - \sqrt{7}) = 8 + 2 \sqrt{7}, \qquad w = 12 - 2(5-\sqrt{7}) = 2 + 2 \sqrt{7}.$