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10 2 ×(10 - 6 ) 3 =10 - 16 km 3 . This means, the volume of all of the grains is about 10 19 ×10 - 16 km = 10 3 km 3 , which is similar in size to our fictitious Mt. Ever est.

1 mile height

inches, so the

of

the

pile

would be miles high. This is an

example of exponential growth .

The Chessboard Activity

This problem is well - known and could be replicat ed, initially, with a plentiful supply of M&Ms us ing a large - scale chessboard. Typically, students engage in this activity using rice or wheat grains. The goal is to place one grain on the first square, two grains on the second, four grains on the third, and so on. They are doubling the number of grains on each subsequent square. They are asked, how many grains of wheat would be on the last square on the chessboard? The following table illustrates this doubling progression of wheat grains on each square on the chessboard.

We can also consider the number of wheat grains located on all or any of the previous squares, that is, the first p squares by summing the number on each previous square. We find a counter - intuitive result when we examine the characteristics for powers of 2 more closely. Below we show the rela tionship between the former sums of the power of two and the current power of two. The counter intuitive result demonstrates a unique property of exponential functions.

Table 1 : Number of wheat grains on chess board squares

(1)

Now double S p .

The students quickly find how difficult it is to con tinue with this process. They turn to mathematics to find the number of grains on the 64th square, which is, 2 63 = 9,223,372,036,854,775,808 grains, or close to 10 19 grains of wheat. In words, this is 10 million trillion or 10 quintillion. How much is that in terms of volume? Let us try to make a com parison with something we know that is really massive, the mountain Mt. Everest. We will model the mountain using a geometric shape of a cone. Let us suppose our fictitious Mt. Everest has a height and radius of 10 km. This means, the vol ume is about 10 3 km 3 . Let us also suppose that a large grain of wheat has a rectangular shape with measurements, 10 mm long and 3 mm square on the base. This implies the volume is ≈ 10 2 mm 3 or

(2a)

(2) – (1) implies

(2b)

This means that the increase found on any dou bling, exceeds the sum of all the previous doubling or growth before it. Furthermore, this result applies to continuous growth situations (see Appendix).

Bacteria in a bottle: The end of their world

Suppose a single bacterium is placed in an empty water bottle at noon and divides the one bacteria into two daughter bacteria after one minute. After 2 minutes, the two bacteria would each divide into two daughter bacteria to result in 4 bacteria, after 3 minutes there are 8 bacteria. This doubling process

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