Chocolate cake

7465 /www/nrich/html/content/id/7465/ 1 2011-06-13T15:49:06 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <ul id="stemLinks"> <li><a href="http://nrich.maths.org/89">Warm-up</a></li> <li><a href="http://nrich.maths.org/7466">Try this next</a></li> <li><a href="http://nrich.maths.org/7393">Think higher</a></li> <li><a href="http://nrich.maths.org/1408">Read: mathematics</a></li> <li><a href="http://www.deliaonline.com/how-to-cook/baking/all-about-cake-tins.html">Read: science</a></li> <li><a href="http://www.deliaonline.com/how-to-cook/baking/the-science-of-cake-making.html">Explore further</a></li> </ul> <div> </div> 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?</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why do this problem?</h3> <a href="https://nrich.maths.org/7465">This problem</a> 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. <h3>Key questions</h3> 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? <h3>Possible extension</h3> 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? <h3>Possible support</h3> <a href="http://nrich.maths.org/89">Making Boxes</a>, which is a Stage 2 problem, would be a good warm-up problem.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">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! </mdoxml> 2 3 0 0 1 0 0 Chocolate cake If I don't have the size of cake tin specified in my recipe, will the size I do have be OK? Measures and Mensuration Volume and capacity Admin Individual Applications Food technology Applications Maths Supporting SET Applications STEM - design technology