Optimization: the biggest rectangle for a fixed perimeter

An animation shows a rectangle with fixed perimeter 20 changing shape as its width varies, tracing the area function A(w) = w(10 - w) on a graph. The derivative 10 - 2w is shown crossing zero at w = 5, marking the maximum area of 25, achieved by a square. Useful for students learning optimization, connecting algebraic functions, calculus derivatives, and geometric intuition about shape and area.

16:9 · every frame verified for overlaps, spacing and edges before rendering

The prompt that made it

With perimeter 20, the rectangle's area A(w) = w(10 - w) is traced as the width changes; the derivative 10 - 2w = 0 gives w = 5 and the square, area 25, wins.

Make your own version

Make the next one in this series

• Show the same optimization for fixed area minimizing perimeter instead.
• Extend to a rectangle with one side against a wall, changing the constraint equation.
• Compare optimization using derivatives versus completing the square for the same problem.

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