Example 3 (continued)

A smidgeon of arithmetic, on the left, and voilà (our initial answer appears as)...

0.0435 in	~	E-sc.rad

However, this is not exactly a very satisfying answer other than the realization that the scaled radius is quite small.  Just how small is 0.0435 inches?  This provides the motivation for the second part of the problem.  To get at that answer, we need to think of a round or spherical object whose radius is close to our answer.  It is perhaps more reasonable to think in terms of the diameter, and so let us double our original (radius) answer to obtain 0.087 inches.  To many of us this is still not an easily identifiable measurement, therefore we shall take matters one step further by converting it to millimeters.  Now, knowing that there are 2.54 cm per inch, it should not be a quantum leap to recognize that there are also 25.4 mm per inch and this is a most convenient conversion fraction to deploy in our quest: This latter result informs us of the need to imagine something with a size of about 2 mm.  A relatively common object that fits such a description would be a "BB."  Or if you tear a small corner piece of letter-size paper and wad it

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into a tiny ball this would also approximate the the size of the Earth when scaled down by the same factor that transforms the Sun into a basketball... QED

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* * *      Another proportion problem we encounter is a rather intriguing use of scale for understanding a time-related matter.  Millions and billions of years are terms that we often hear when either astronomical or geological phenomenon relating to our very own Earth are discussed.  To help us grasp, at least crudely, the magnitude of such large numbers we will "scale" down the age of the Earth to exactly one "calendar" year...  

Example 4

Most scientists agree that the Earth's age is approximately four and one-half billion years old, while the amount of time humans (or hominids) have been around is roughly about four and one-half million years.  If the length of time that Earth has existed is compressed into a 365-day, calendar year where January 1st marks the formation of the Earth, then on which day of the year would human beings have begun to appear on the planet?

      First, let us focus on how many days would correspond to 4,500,000 years using a ratio (i.e.,