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# Efficient Cutting

**Can you work out some possible dimensions of a rectangle and two circles which can be cut from a single sheet of A4 paper and put together to make a cylinder?**

Which dimensions allow you to make a cylinder with the greatest volume?
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### Plutarch's Boxes

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*Efficient Cutting printable sheet*

Cylindrical containers, like the tin cans used to package some food, can be made by using two circles for the ends, and a rectangle which wraps round to form the body.

To make cylinders of varying sizes, the three pieces can be cut from a single rectangle of flat sheet in several ways.

For example:

Which dimensions allow you to make a cylinder with the greatest volume?

*You can assume that the dimensions of an A4 sheet of paper are 21cm and 29.6cm*

Click here for a poster of this problem.

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal spoons. Each day a spoonful was used to perfume the bath of a beautiful princess. For how many days did the whole jar last? The genie's master replied: Five hundred and ninety five days. What three numbers do the genie's words granid, ozvik and vaswik stand for?