More Optimization — Question 1

Question 1 - Solution

Let $x$ be the length of the square cut from each corner.

After folding, the dimensions of the box will be:

Height = $x$

Length and width = $24 - 2x$

Volume: $$V(x) = x(24 - 2x)^2$$

Step 1: Expand volume function

$$V(x) = x(576 - 96x + 4x^2) = 576x - 96x^2 + 4x^3$$

Step 2: Find critical points

$$V'(x) = 576 - 192x + 12x^2 \Rightarrow 12x^2 - 192x + 576 = 0 \Rightarrow x^2 - 16x + 48 = 0 \Rightarrow x = 8 \pm \sqrt{16} \Rightarrow x = 8 \pm 4$$

$$x = 4 \quad \text{or} \quad x = 12$$

But $x = 12$ makes $24 - 2x = 0$ → not valid

Valid critical point: $x = 4$

Answer: Cut out squares of size $\boxed{4 \text{ in}}$ to maximize volume