50
Summary
Notation
........................................
is proportional to
~........................................on the order of, is on the order of
Summary
Nature behaves differently on large and small scales. Galileo showed that this results fundamentally from
the way area and volume scale. Area scales as the second power of length, A
.
L
2
, while volume scales as
length to the third power, V
.
L
3
.
An order of magnitude estimate is one in which we do not attempt or expect an exact answer. The main
reason why the uninitiated have trouble with order-of-magnitude estimates is that the human brain does not
intuitively make accurate estimates of area and volume. Estimates of area and volume should be approached
by first estimating linear dimensions, which one’s brain has a feel for.
Homework Problems
1 . How many cubic inches are there in a cubic foot. The answer is not
12.
2. Assume a dog's brain is twice is great in diameter as a cat's, but each
animal's brain cells are the same size and their brains are the same shape. In
addition to being a far better companion and much nicer to come home to,
how many times more brain cells does a dog have than a cat. The answer is
not 2.
3 . The population density of Los Angeles is about 4000 people/km
2
.
That of San Francisco is about 6000 people/km
2
. How many times farther
away is the average person's nearest neighbor in LA than in San Francisco.
The answer is not 1.5.
4. A hunting dog's nose has about 10 square inches of active surface. How
is this possible, since the dog's nose is only about 1 in x 1 in x 1 in = 1 in
3
.
After all, 10 is greater than 1, so how can it fit.
5. Estimate the number of blades of grass on a football field.
6. In a computer memory chip, each bit of information (a 0 or a 1) is stored
in a single tiny circuit etched onto the surface of a silicon chip. The circuits
cover the surface of the chip like lots in a housing development. A typical
chip stores 64 Mb (megabytes) of data, where a byte is 8 bits. Estimate (a)
the area of each circuit, and (b) its linear size.
7. Suppose someone built a gigantic apartment building, measuring 10 km
x 10 km at the base. Estimate how tall the building would have to be to
have space in it for the entire world's population to live.
8. A hamburger chain advertises that it has sold 10 billion Bongo Burgers.
Estimate the total mass of feed required to raise the cows used to make the
burgers.
9. Estimate the volume of a human body, in cm
3
.
10 S. How many cm
2
is 1 mm
2
.
SA solution is given in the back of the book.A difficult problem.
A computerized answer check is available.
.
A problem that requires calculus.
Chapter 1Scaling and Order-of-Magnitude Estimates