### Boxed In

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

### Plutarch's Boxes

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:

The recipe uses a tin of radius 10cm and depth 7.5cm. This has a volume of

$$\pi\times10^2\times7.5 cm^3 = 2400 cm^3$$

We don't know the depth of the 23cm round tin.  If its depth is also 7.5cm, then its volume is given by:

$$\pi\times11.5^2\times7.5 cm^3 = 3100 cm^3$$

which would be fine.

The limit for the depth of this tin could be found by trial and error, or you could rearrange the formula for the volume of a cylindrical tin to find the height which gives a volume of 2400 cm3.

$$\pi\times11.5^2\times h cm^3 = 2400 cm^3$$ $$h = \frac{2400}{\pi\times11.5^2} cm = 5.8 cm$$

So depending on the depth of the 23cm round tin, all could be well and Toby gets his cake!

The volume of the square tin is $15^2\times6cm^3=1350cm^3$, which isn't large enough.  For a large enough square tin of the same depth, we need:

$$l^2\times 6 cm^3 = 2400 cm^3$$
$$l = \sqrt{\frac{2400}{6}} cm = 20 cm$$

where l is the length of the side of the tin.

If you think of the diagonal of the square tin as being equivalent to the diameter of a round tin, the length of the diagonal, d, is:

$$d^2 = 2\times l^2$$
$$d = \sqrt{2\times20^2} = 28.3 cm$$

so quite a bit longer than the diameter of the 20cm round tin!