Maximum Area Fence Problem

Calculus 1 · Axiom Academy

A farmer has 100 feet of fencing and one question — what shape of pen holds the most? Put the optimization in your hands. You've got exactly 100 ft of fence for a rectangular pen. Spend it all, and one shape encloses more land than any other — here's how to find it, three ways. All 100 ft of fence goes around the pen, so widening it must shorten it: length is always 50 − width . Drag the width and watch the enclosed area swell, peak, then shrink. Write the area as one function of the width: A(w) = w(50 − w) = 50w − w² . Its graph is a hill. Drag the width and watch the tangent line — the peak is exactly where that line goes flat, where the slope A (w) = 50 − 2w hits zero. How much are you leaving on the table? The math says the best pen is the 25 × 25 square at 625 ft² . So how much land does any other shape cost you? Drag your width and watch the gap to the square — and notice it's symmetric: a long-thin pen loses just as much as a short-wide one. One fence, one move that finds the best of anything: write the quantity as a function, then set its derivative to zero . Here it crowns the square — but the same step sizes a shipping box for least cardboard, a fold for the strongest beam, a price for the most profit. Fixed perimeter? The square wins.

This is the written version of the interactive lesson above. See the full Calculus 1 course.