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A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

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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?

The Genie in the Jar

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?

Chocolate Cake

Age 11 to 14
Challenge Level

Why do this problem?

This problem gives practice in working with volume in a context which anyone who bakes will recognise - when your recipe specifies a piece of equipment you don't have.  Rather than just going out and buying another tin, you therefore want to see if what you have will do.

The depth of the 23cm round tin is deliberately omitted, since depth is rarely specified in a recipe in fact.  Students will need to consider what depth such a tin might realistically have, and then see if they think the volume of cake mix will fit in it.

Key questions

What is a realistic depth for the 23cm round tin?

What is the least depth the 23cm round tin could have, and still be suitable to bake the cake?

Possible extension

Once students have explored the question asked, they could then consider whether a round tin or a square one takes a bigger volume of cake mix for a given diameter.  What's the equivalent of diameter in a square tin, is it the side length or the diagonal?

Possible support

Making Boxes, which is a Stage 2 problem, would be a good warm-up problem.