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Optimization

Cut squares from the corners of a sheet and fold it into an open box. The biggest box sits exactly where the volume curve's tangent goes flat.

Step 1 of 14

The sheet, from above

Folded into a box

Speed

A 30 cm square sheet of card (3 dm). Cut an x-by-x square from each corner and fold up the four flaps to make an open box. Cut too little and the box is shallow; cut too much and the base vanishes. Which cut holds the most?

V(0.05) = 0.05\,(3 - 2 \cdot 0.05)^2 = 0.05 \times 2.9^2 = 0.421 \text{ litres}

V'(0.05) = (3 - 2 \cdot 0.05)(3 - 6 \cdot 0.05) = (2.9)(2.7) = 7.83

V(x) = x(L − 2x)². By the product and chain rules V′(x) = (L − 2x)² − 4x(L − 2x) = (L − 2x)(L − 6x), which is zero at x = L/6.

Optimization

Cut squares from the corners of a sheet and fold it into an open box. The biggest box sits exactly where the volume curve's tangent goes flat.

Step 1 of 14

The sheet, from above

Folded into a box

Speed

V(0.05) = 0.05\,(3 - 2 \cdot 0.05)^2 = 0.05 \times 2.9^2 = 0.421 \text{ litres}

V'(0.05) = (3 - 2 \cdot 0.05)(3 - 6 \cdot 0.05) = (2.9)(2.7) = 7.83

V(x) = x(L − 2x)². By the product and chain rules V′(x) = (L − 2x)² − 4x(L − 2x) = (L − 2x)(L − 6x), which is zero at x = L/6.

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Optimization

See also

Optimization

See also