Download Scaling Lecture 3: Estimation

Scaling Lecture 3: Estimation
Order of Magnitude: a very rough estimate of a value, accurate to within one power of
ten. The numbers 1, 2, 3 and 8 are all the same “to one order of magnitude”.
Likewise the numbers 10,000, 16,750, 9,767 are all the same to one order of
magnitude, or “have the same order of magnitude”. But 3 and 10,000 are obviously
different. You could also argue that 1 and 8 are getting close to different, because 8
rounds to 10, and 1 and 10 are different by one order or magnitude.
Give answers to the following, correct to one order of magnitude.
How tall are you in meters?
Even if you don't know how to convert feet and inches to cm, you can answer this.
You are about 1 m tall (or more accurately 2m). You are obviously not 0.1 m tall or 10 m
tall, so 1 m tall is the correct answer to one order of magnitude.
How many square meters per person when our classroom is full?
Again you might be wildly wrong on your estimates of how many meters big the
room is, but you probably won't be wrong by 10 times! Say the room is 10m by 20m and
there are about 100 people in the room. We get 200/100 = 2 meters squared per person.
How big is a cell?
You might not have any idea here. Although cells have vastly different sizes, a
rough idea would be about 10 microns, or 10­5 m in diameter. If you want to estimate unknown quantities to within one order of magnitude, you can use
common sense (as illustrated above), plus some fundamental ideas of size.
eg. Use the size of the cell to estimated above to guess the number of cells in your brain.
Let's say the cell is like a little cube. Then the volume of one cell is
10­5 m x 10­5 m x 10­5 m = 10­15 m3
What's the volume of your brain? Just looking at your head you can guess:
10cm x 10cm x 10cm = 4 x 10­6 m3
Total number of cells in brain = total volume/volume per cell = 4 x 109
How did we do? We can look up that the adult human brain has about 100 billion
neurons. That's 108 neurons. But there are actually 10 to 50 times as many “glial” cells,
the supporting cells for the neurons. So in total, neurons + glial cells would be about
109 to 5 x 109 cells. Our ballpark estimate was pretty good!
Sizes to know for this course:
human hair, dustmite
cells, pollen, yeast
bacteria
virus
individual molecule
100 microns
10 microns
1 micron
10­100 nm
Angstroms
10­4 10­5 10­6 10­7 to 10­8 10­10 Again, these are very rough order of magnitude sizes.
Practice problems; answer to one order of magnitude accuracy:
Bacterial colonies are visible on your agar as tiny patches, about 1 mm across.
Assume that bacteria grow in a single layer. How many bacteria per colony?
(answer: 1 million bacteria per colony)
What volume of air is available per person in our classroom?
(answer: maybe 5­6 meters cubed per person)
What's the maximum number of hairs you could have on your head?
(answer: each hair takes up maybe 10­4 x 10­4 = 10­8 square meters.
Surface area of a sphere is 4 PI r2, call that 12 r2.
The radius of your head is on the order of 10 cm (not 1 cm and not 100 cm!).
Surface area of your head is therefore roughly 12x10­2 square meters.
So we could fit 12 million hairs on your head if they were packed tightly. Maybe
divide that by 2 since most of us only have hair on about half our heads.
(In reality we each have more like 100,000 hairs on our heads – not packed maximally))