Quandaries
and Queries 

Name: Dave Who is asking: Parent Question: If I have a known surface area and volume of an unknown object and I want to double the size of the object, how do I find the new area and volume? 

Hi Dave, This is actually a very interesting question and makes an interesting comparison. Let me give a couple applications  from mathematical biology. Consider different sizes of whales (with roughly the 'same shape').
We will see below that volume increases faster than surface area. Therefore, if you scale up the 'whale shape' you find that the amount of oxygen stored increases faster than the oxygen leaving the blood. Therefore, a larger whale can dive for a longer time. Therefore, if you scale up the whale shape, you will find that the energy to beat the heart (and the blood flow needed) goes up slower than the amount of blood pumped per heart beat. Therefore, the heart rate slows down. This slowing of heart beat runs across many species. A humming bird has a very fast heart beat  an eagle or condor a slower heart beat. Now to the rate at which things change. You can test
it for a cube:
Notice that the volume grows faster than the surface area Grown in surface area, for doubling: 24/6 = 2400 /600 = 4 These factors are a constant for 'scaling' up all volumes and surface areas. If you scale up by a factor of K, the volume goes up as K^{3} and the surface area goes up by K^{2}. (Length goes up by K) Here is a second situation where this 'scaling' matters. If you change units  from meters to centimeters. The areas go up from square meters to square centimeters  by 1000^{2}. The volume goes up from cubic meters to cubic centimeters  by 1000^{3}. I am not sure whether I can give a simple 'proof' of this scaling. There is one, if you go into how we actually 'measure' such quantities. In calculus, this is really done by chopping volumes up into small cubes  and adding up the number of cubes. Therefore the analysis for cubes just applies to the pieces and we have the same number of 'larger' pieces. Something similar happens for surface area. Once you know that it does not depend on WHICH shape, you can just use the simplest shape  the cube. In closing, you might be able to apply this reasoning to explain WHY there is a limit to the size of bird which can fly, using the standard techniques birds use. (The energy needed is proportional to the weight  the volume.) Walter 

