# Chocolate cake

Toby is 2 tomorrow and he wants a big sticky chocolate cake for his party! The problem is that my recipe specifies a 20cm round tin with a depth of 7.5cm, but I haven't got a 20cm round tin.

I do have a 23cm round tin, and I've also got a square tin, which has a side length of 15cm and a depth of 6 cm.

Which one do you think would be best? Or should I go out and buy a new cake tin?

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 cm^{3}.

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

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!

### 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?