... emerges. In each case the optimum cutout size is 1/6 of the size of the square sheet i.e.: A square that is 24cm long has an optimum cutout size of 4cm and a maximum volume of 1024cm3 A square that is 120cm long has an optimum cutout size of 20cm and a maximum volume of 128000cm3 Proof A more mathematical approach can also be used to accurately determine the optimum cutout size, using a combination of quadratics and calculus to achieve a general result. The volume of the box (y) is calculated by: length * width * height =(w - 2x)*(w - 2x)*x (where w is the length of the box) =x(4x2 - 4xw + w2) =4x3 - 4x2w + w2x (plotting this equation against y will display volume against cutout size so long as 0

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