 Quandaries and Queries

Name: Dave

Who is asking: Parent
Level: Middle

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').

1. The amount of oxygen it holds during a dive is proportional to the volume of the lungs;
2. The amount of oxygen used in the muscles is proportional to the surface area of the blood vessels (how it gets from the blood to the muscles).
3. the amount of blood pumped for each beat of the heart is proportional to the volume of the heart.
4. the amount of energy to beat the heart is proportional to the surface area of the blood vessels in the heart.

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:

Cube dimensions Volume Surface area
(6 squares)
1 1 1 1 1 1 = 1 6 1 1 = 6
2 2 2 2 2 2 = 8 6 2 2 = 24
10 10 10 10 10 10 = 1000 6 10 10 = 600
20 20 20 20 20 20 = 8000 6 20 20 = 2400

Notice that the volume grows faster than the surface area

Growth in volume, for doubling: 8/1 = 8000/1000 = 8

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 K3 and the surface area goes up by K2. (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 10002. The volume goes up from cubic meters to cubic centimeters - by 10003.

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

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