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Maximum Box Investigation Question: You have a square sheet of card 24cm by 24cm. You can make a box (without a lid) by cutting squares from the corners and folding up the sides. What size corners should you cut so that the volume of the box is as large as possible? The first step to evaluating the maximum possible volume that the box could contain is to find the area of the paper. This is simply 24cm x 24cm = 576cm². Now that we have the area, it is easy to evaluate the maximum volume in a variety of different shapes. First of all, I'm going to try a perfect cube, which i predict to be the best shape. With no lid, the cube has 5 sides, so the 576cm² must be divided by 5. This gives us 115.2cm² for each of the cube's faces. The square root of 115.2cm² is 10.73cm, and...

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