Elephant Feet

Date: 05/16/97 at 00:56:47
From: David Beecham
Subject: Volume and surface area

Why do elephants have big feet? 
 
We've tried very hard to start answering this question but have had no 
luck. Could you please get us started? 

Date: 05/16/97 at 06:00:51
From: Doctor Mitteldorf
Subject: Re: Volume and surface area

Dear David,

The more area the bottom of a foot has, the less pressure on each 
square centimeter.  I don't know exactly how elephants walk, but 
periodically each foot must bear a significant fraction of the total 
mass of the elephant.

The strength of a bone or any body part is proportional to its surface 
area. If you take two bones made of the same material and try to break 
them, then it will be four times as hard to break one that's twice the 
diameter.

On the other hand, the bone (or any body part) becomes heavier when
you make it bigger in proportion to its volume. If you make the bone 
twice the diameter, and also twice the length, then its volume is 
2*2*2 = 8 times as big. That means it's 8 times as heavy.

Suppose you're designing a dog that's 1 foot tall and weighs 10 
pounds. You adjust the proportions so that the bones are strong enough 
to hold him up without breaking (even when he's running fast).  

Now try to apply the same design to an elephant that's 10 feet tall 
and weighs 10,000 pounds. If you kept all the proportions the same, 
but made the dog 10 times as high, 10 times as wide, 10 times as long, 
he would weigh 1000 times as much (that's where I got the number 
10,000 pounds). But his legs would only be 100 times as strong 
(because the diameter of all the bones and muscles would be 10 times 
as big). This would be a very weak creature - unable to walk, with 
bones always liable to breakage.

The upshot is: it's easier to design a small animal to be strong 
(ants can lift 100 times their weight). It's hard to design a large 
animal to be strong (an elephant may weigh 10,000 pounds but it can 
only carry 2,000 pounds extra).  

Elephant muscles and bones need extra reinforcement, and the 
proportions of the animal are adjusted to help carry the weight.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/