Now let's take a critical look at our model. We decided to model the earth as
a perfect sphere of radius 4000 miles with a population of 5.5 billion people.
What problems are there with this model?
Click here for the answer
We obviously need to develop a better mathematical model. There are a number
of things that we could try, but perhaps we should simply return to a reference
book. According to the 1991 Rand McNally World Atlas:
- The earth's population (at some time) in 1990 was 5,236,000,000.
- The total land area of the earth, including inland water and Antarctica,
is 57,800,000 square miles.
Based upon these numbers, let's repeat the square foot calculation. With a new
model, we now need a new method that recognizes the uncertainty in the numbers
given above. Although the atlas doesn't say so, let's assume that the
population is rounded off to four significant digits and that the area is
rounded off to three significant digits. This means that no more than the
first three digits of our ultimate answer can be considered significant.
By using the improved figures and doing some division, we should be able to
calculate the number of square feet of land per person at some time in 1990.
See if you can do the calculation yourself.
Click here for the answer
Rounding our results to three significant digits (since the original land area
estimate was significant only to three digits), we obtain that at some time in
1990, there were approximately 308,000 square miles per person.
Are the first three digits in the answer really significant? We can
answer this question with a technique known as interval arithmetic.
Based upon our assumptions about the significance of the two measurements, we
know that the true land surface area is between 57,750,000 and 57,850,000
square miles and that the true population is between 5,235,500,000 and
5,236,500,000 people. (Be sure that you understand why.) We can calculate an
upper and a lower bound on the true answer by doing two calculations.
We obtain an upper bound on area per person by dividing the large
value for the area by the small value for the population:
| 5.785e7 * 5280^2 / 5.235e9; |
Similarly, we obtain a lower bound on area per person by dividing the
small value for the area by the large value for the population:
| 5.775e7 * 5280^2 / 5.2365e9; |
We thus see that there really weren't three significant digits in the answer.
The true answer is somewhere between 307,452.9934 and 308,044.2059 square feet
per person. We can leave it at that, or we can say (less precisely but more
succinctly) that the area per person, rounded to two significant digits, is
310,000 square feet. We can say this because the upper and lower bounds agree
on the first two digits.
Joseph L. Zachary
Hamlet Project
Department of Computer Science
University of Utah