A box has faces with areas 3, 12 and 25 square centimetres. What is
the volume of the box?
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?
From a square sheet of paper 20cm by 20cm, we can make a box without a lid by cutting a square from each corner and folding up the flaps.
Find the different volumes of box it is possible to make by varying the size of the cut-out squares.
What is the maximum possible volume?
What is the size of the square cut-out that produces it?
Now try starting with different sized square sheets of paper.
Can you find a relationship between the size of paper and the size of the square cut-out that produces the maximum volume?
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