Sizu Riatto
Calculus · Chapter 13

Best Shape Finder

Draw it, name the letters, write the thing to make best, use the limit to leave one letter, set its slope to zero, and check the ends.

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Opens on Chapter 13’s example: Pilar’s 120 feet of fence by the river; a 20 by 20 inch sheet of cardboard; a 355 mL (12 fl oz) can; a poster with 96 square inches of print.

1. A fenced rectangle
2. An open box from a sheet
3. A closed can
4. A poster with margins
5. A pipe: along the road, then across the field
Enter your numbers, then press Work it out.

How it works

For the fence, if each side away from the river is x, the long side is the rest of the fence, so the area is x × (length − 2x); its slope is zero at a quarter of the fence. With all four sides fenced, the best rectangle is a square. For the box, cutting x from each corner of a w × h sheet leaves a box x tall with volume x(w − 2x)(h − 2x); the best cut is where its slope is zero (and between 0 and half the shorter side). For the can, fix the volume, write the metal as the two ends plus the side, and the slope is zero when the height equals the diameter. Real cans are taller, for reasons the model leaves out, such as thicker ends.

From the book, Chapter 13. By the river: 30 by 60 feet, 1,800 square feet. The 20-inch sheet: cut 3.33 inches for 592.6 cubic inches. The can: r = 1.51 in and h = 3.02 in, 43.0 square inches of metal. A 100-foot rope: 25 by 25, 625 square feet. Every chapter and its calculator →

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