Maximizing the Volume of an Open‑Top Box from a Square Sheet

Overview

A square aluminum sheet of side $a$ has its four corners cut out as squares of side $x$. The remaining flaps are folded up to form an open‑top box. The animation derives the volume function $V(x)$, finds its maximum, and displays the optimal cut size and the maximal volume.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Show a flat square sheet, highlight the four corner squares of side $x$ being cut, then fold to reveal the open box.
2	Derive Volume	~5s	Animate the dimensions of the box (length $a-2x$, width $a-2x$, height $x$) and write the volume formula $V(x) = x(a-2x)^2$.
3	Optimize	~7s	Plot the cubic function $V(x)$ on a coordinate axis, draw the derivative $V'(x)$, locate the critical point where $V'(x)=0$, and highlight the feasible interval $0 < x < a/2$.
4	Result	~4s	Show the optimal cut size $x = \frac{a}{6}$ and the maximal volume $V_{\max}=\frac{2a^3}{27}$. Conclude with a brief visual of the box at this optimal size.
5	Outro	~1s	Fade out to a simple caption summarizing the key takeaway.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│                MAIN (visuals)               │
│   – square sheet, cuts, folding, graphs –    │
│                                             │
├─────────────────────────────────────────────┤
│ Caption / key formula (small persistent text)│
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Animated square sheet, cut squares, folding into box; algebraic derivation; graph of $V(x)$ and its derivative; highlight of optimal point.	Central visual focus throughout all phases.
Caption	Current key formula (e.g., $V(x)=x(a-2x)^2$ or $x=\frac{a}{6}$) displayed in a small footer.	Updated at the start of each new phase; omitted during pure visual actions if not needed.

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Notes

• Assumptions: The side length $a$ is treated as a positive constant; the animation uses a generic symbolic $a$ rather than a specific numeric value. The feasible domain for $x$ is $0 < x < a/2$.
• The derivative is shown by animating a tangent line or by plotting $V'(x)$ alongside $V(x)$.
• No textual narration is required; all information is conveyed visually via equations and highlights.
• Total runtime is kept under 20 seconds to maintain brevity while fully covering the derivation and optimization.