Table of Contents

• Prequiz...
• The Power Function
• 1: Introduction
• 2: What are scaling relationships?
• 3: The power function
• 4: The shape of the power function
• 5: The conversion factor "a"
• 6: The power factor "b"
• Log Transformations
• 7: Introduction
• 8: What are logarithms again?
• 9: Graphing on a log scale
• 10: Visualizing the data
• 11: Getting to a straight line
• 12: The key parameter is "b"
• The value of "b"
• 13: Introduction
• 14: A geometric approach
• 15: Scaling surface area of cubes
• 16: Scaling surface area of cubes
• 17: A "2/3 scaling law"?
• Confronting ideas with data
• 18: Introduction
• 19: An unexpected result
• 20: A "3/4 scaling law"?
• 21: Internal transport systems
• 22: Branching and supply networks
• 23: Deriving the "3/4 scaling law"
• 24: So what's the real value of "b"?
• Review
• 25: Review of Concepts

The Parameters of the Power Function

So, what roles do "a" and "b" play in this equation?

Metabolic Rate = a*(size)^b

Remember, in this function, metabolic rate and size are variables, meaning that they take on a range of values (which is why they are illustrated on the axes of a graph). On the other hand, "a" and "b" are parameters - and in any given situation, they are constants. For example, say you write a dissertation on the relationship between size and metabolic rate in ewoks. You might find the following relationship: For instance, you might find out that the metabolic rate of ewoks is related to their size by the following equation:

Metabolic rate of ewok = 3.69*(size of ewok)^0.98

Notice that "a" and "b" have been replaced with the actual numbers that specify the relationship. In this case, through our intensive study of ewoks, we figured out the relationship between their weight and their metabolic rate, and so now we know the fixed values of those two parameters. If we studied another group of organisms (say, wookies), we would likely find that the parameters were different. For instance, we may find that this relationship turns out to be:

Metabolic rate of wookie = 3.46*(size of wookie)^0.95

We may hypothesize that the numbers are so similar because it appears that wookies and ewoks may be closely related.

	vs.
ewok M.R. ewok = 3.69*(size of ewok)0.98		wookie M. R. wookie = 3.46*(size of wookie)0.95

Now lets think about what range of values "a" and "b" could take on generally, and how different values will change the behavior or shape of our function.

We already know that size is always going to be a non-zero, positive number. And we also know that "a" and "b" can only take on values that result in the metabolic rate also being a non-zero, positive number. Keeping that in mind, lets first think about the value of the parameter "a".