# The scaling tool

#### Prerequisites

- The idea of algebra: Unknowns and relationships
- Functions
- The dimensional analysis tool
- Units, Scaling, and functional dependence

The idea of algebraic symbols and functional dependence has been discussed in the pre-requisites, and it provides one of your most important tools for making sense of and solving problems.

How a parameter or variable comes into an equation tells you how important it is and how it relates to other variables. (See the page, Units, Scaling, and functional dependence.)

For example, we know that the heat generated by a warm blooded animal is proportional to the amount of living mass it has, while the heat lost by that object through radiation (mostly infrared) is proportional to the animal's surface area.

If $L$ is a measure of the animal's size, we know that its volume is proportional to $L^3$ while its surface is proportional to $L^2$. Therefore, as animals get larger, their volume grows faster than their surface area.

This means that smaller animals have to worry about losing too much heat while larger animals have to worry about not losing enough. This affects structure in many interesting ways.

The tool icon I have chosen for functional dependence is a baby elephant and mother representing the scaling up of a system to a larger size.

Here's an example of how to apply this tool.

One of the essential elements of the animal immune system is the macrophage: a cell that ingests and destroys harmful bacteria. The bacterium might contain molecules of a chemical safe for it, but harmful to the macrophage. But the macrophage is bigger, so the density of the harmful molecules will be less. Assume they are both spherical.

Suppose the macrophage has a diameter of 20 μm and the bacterium has a diameter of 1 μm. If the density of these molecules in the bacterium is D, what will be their approximate density in the macrophage once the bacterium has been ingested, broken up, and distributed throughout the macrophage?

1. D

2. D/20

3. D/400

4. D/8000

5. Something else

Since the volume of the cell is proportional to the cube of its dimensions, we expect the volume of the macrophage to be 20x20x20 times as big at the volume of the bacterium or 8000. Therefore, the dangerous chemical will be diluted by a factor of about 8000. Note that all this assumes is that the shapes of the bacterium and the macrophage are similar. We don't have to know exactly what shape they are. And if they differ a little (say one is spherical and the other nearly cubic) our answer will still be pretty good. (For practice, compare the volume of a cube of side 20*D* with the volume of a sphere of diameter *D*.)

Joe Redish 12/24/21

Last Modified: December 24, 2021