Energy and Human Ambitions on a Finite Planet, 2021a

1 Exponential Growth 3
We will see how the rule of 70 arises mathematically later in this chapter.
But first, it is more important to underst<strong>and</strong> the consequences.Tomakethe
math simple, let’s say that a town’s size doubles every 10 years (which
by the rule of 70 corresponds to a 7% growth rate, incidentally). Starting
in the year 1900 at 100 residents, we expect town population to be 200
in 1910, 400 in 1920, 800 in 1930, eventually climbing to over 100,000
by the year 2000 (see Table 1.1). Unabated 7% growth would result in
the town reaching the current world population just 260 years after the
experiment began.
But let’s explore an example that often reveals our faulty intuition around
exponential growth. Here, we imagine a jar rich in resources, seeded
with just the right number of bacteria so that if each bacterium splits
every 10 minutes, the jar will become full of bacteria in exactly 24 hours.
The experiment starts right at midnight. The question is: at what time
will the jar be half full?
Think about this on your own for a minute. Normal intuition might
suggest a half-full jar at noon—halfway along the experiment. But what
happens if we work backwards? The jar is full at midnight, <strong>and</strong> doubles
every ten minutes. So what time is it half full?
Table 1.1: Example 7% growth progression.
Year
Population
1900 100
1910 200
1920 400
1930 800
1940 1,600
.
.
2000 102,400
10 minutes is perhaps a little fast for biology,
but we’re looking for easy underst<strong>and</strong>ing
<strong>and</strong> picking convenient numbers. In practice,
20–30 minutes may be more realistic.
We will also ignore deaths for this “toy” example,
although the net effect only changes
the rate <strong>and</strong> not the overall behavior.
The answer is one doubling-time before midnight, or 11:50 PM. Figure
1.1 illustrates the story. At 11 PM, the jar is at one-64 th capacity, or 1.7%
full. So, for the first 23 of 24 hours, the jar looks basically empty. All the
action happens at the end, in dramatic fashion.
1/512 1/256 1/128 1/64 1/32 1/16 1/8 1/4 1/2 1
22:30
22:40
22:50
23:00
23:10
23:20
23:30
23:40
23:50
24:00
Figure 1.1: The last 90 minutes in the sequence of bacteria (green) growing in a jar, doubling every 10 minutes. For the first 22.5 hours,
hardly anything would be visible. Note that the upward rise of green “bars” makes an exponential curve.
Now let’s imagine another illustrative scenario in connection with our
jar of bacteria. The time is 11:30 PM: one-half hour before the end. The
jar is one-eighth full. A thoughtful member of the culture projects the
future <strong>and</strong> decides that more uninhabited resource-laden jars must
be discovered in short order if the culture is to continue its trajectory.
Imagine for a second the disbelief expressed by probably the vast
majority of other inhabitants: the jar is far from full, <strong>and</strong> has served for
141 generations—a seeming eternity. Nonetheless, this explorer returns
reporting three other equal-sized food-filled jars within easy reach. A
hero’s welcome! How much longer will the culture be able to continue
growing? What’s your answer?
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.