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#197 - Tools Of Our Trade -- Part 1: How To Project Future Growth, 04-Sep-1990

One of the important characteristics of environmental problems is the
way they're growing. It is important for the public (and for news
reporters and environmentalists) to understand growth. This is often
easy to do because of the way most things grow. As Ralph Lapp has made
clear in his book, THE LOGARITHMIC CENTURY, human population (and most
of the things related to humans such as automobiles, chemicals and
chemical wastes), are growing exponentially. This permits us to make
accurate growth projections easily.

First a definition: A quantity is growing exponentially if it grows by
a fixed percentage of the whole in a fixed time period. A familiar
example of a quantity that is growing exponentially is a bank account
that grows at 6% per year; it grows by a fixed percentage of the whole
(6%) in a fixed period of time (a year).

There are some rules about exponential growth that allow us to make
quick and accurate projections into the future.

RULE 1: To determine the doubling-time (d) for an exponentially-growing
quantity, divide the annual percentage rate of increase (p) into 70.

d = 70/p [Rule 1]

where:

d = the time it takes for the quantity to double in size;

p = the annual increase expressed as a percentage.

Thus the savings account growing at 6% per year is doubling every 70/6
= 11.7 years. Thus $5 growing at 6% per year will grow to $10 in 11.7
years. By the same reasoning, a quantity that is growing at 10% per
year--such as production of a chemical--will have a doubled annual
production rate in 7 years. (For those who are curious, 70 is used
because it is very close to 100 times the natural logarithm of 2, which
is 0.693.)

RULE 2: If we know the doubling time for an exponentially growing
quantity we can calculate the annual percentage increase (p) by using a
variation of Rule 1.

p = 70/d [Rule 2]

where:

d = the time it takes for the quantity to double in size;

p = the annual increase expressed as a percentage.

If we are told that something is doubling in 5 years, we know that it
is growing at 70/5 = 14% per year.

RULE 3: The fundamental equation for exponentially growing quantities
is:

N_sub_t = N_sub_o*e**kt [Rule 3]

where:

N_sub_o is some original amount;

N_sub_t is the amount that it has grown to at some later time, t;

e is a constant, equal to 2.718 (it is the base of natural logarithms);

k = the annual percentage increase expressed as a decimal fraction (in
other words, it's the value we've been calling p, divided by 100);

t = time (in any units you care to choose).

Don't be put off by the strange notation; N_sub_o is pronounced "N sub
O" and N_sub_t is pronounced "N sub T." This is the way mathematicians
and physicists like to talk about quantities, but once you get used to
the odd way of expressing them, the ideas themselves are simple enough.
To handle the arithmetic involved in such an equation, remember that
when two items are written next to each other, it means that they
should be multiplied together. In this example, k and t have been
written kt and this means that k is multiplied by t. (We have also used
an asterisk to indicate that two numbers should be multiplied by each
other, so kt and k*t mean the same thing--multiply k times t.)

There is a standard order in which mathematical operations are carried
out. First, any exponents should be evaluated (figured out). In this
case, kt is an exponent, so you multiply k times t first. Next you
carry out the exponentiation: in this case, you raise e to the power of
k*t. (A $15 scientific calculator from Radio Shack can raise e to any
power for you.) Next you carry out any multiplication or division; in
this case, because they are written next to each other, you would
multiply N_sub_o times whatever you got when you raised e to the power
of kt. Last, you do any addition or subtraction; in this particular
example there isn't any addition or subtraction indicated.

Parentheses are used to change the order in which mathematical
operations are carried out; always do what's inside parentheses first.
Start inside the innermost parentheses and work your way outward.

Example of Rule 3: If production of hazardous wastes is growing at 6.5%
per year [thus doubling every 10.8 years] and if we produced 30 million
tons of hazardous waste in 1980, how much hazardous waste will we be
producing in 1995? N_sub_o = 30 million tons; t = 1995-1980, or 15; k =
6.5/100, 0.065. Therefore, N_sub_t (the amount of waste produced at
time t), when t = 15, is e raised to the power of (0.065 x 15, or
0.975), times 30 million. Using a scientific calculator, we raise e to
the power of 0.975 and we get 2.65. Therefore, the amount of waste to
be produced in 1995 = 30 million tons times 2.65, or 79.5 million tons,
assuming that the growth-rate continues to average 6.5% per year
between 1980 and 1995.

RULE 4: If a quantity is growing exponentially, during one human
lifetime (assumed to be 70 years) it will grow by a factor of 2 raised
to the power of p, where p is the annual percentage rate of increase.
(The phrase "it will grow by a factor of" means "its growth can be
calculated by multiplying by.")

N_sub_t after 70 years = N_sub_o*2**p [Rule 4]

where:

N_sub_o is some original amount;

N_sub_t is the amount that it has grown to at some later time, t;

p = the annual increase expressed as a percentage.

Table 1 gives 2p for many typical values of p.

Thus when we say that production of chemical X is increasing at 10% per
year, we can calculate that during one human lifetime the annual
production rate of chemical X will increase by a factor of 2**10, or
1024. That is to say, if we produced 1,000,000 (one million) pounds of
chemical X in 1980 and our production is growing at 10% per year, at
the end of one human lifetime we will be producing 1,000,000 x 1024 =
1,024,000,000 (or more than one billion) pounds of chemical X annually.

At this point we should make the distinction between predictions and
projections. A prediction is a statement of what someone thinks is
going to happen. A projection is a statement of what will happen if
things don't change. As we are using the term here, a projection is
based only on the past record of the growth of something. A prediction
may take into consideration many other factors besides the past record
of the growth of something; for example, a prediction may take into
account how we humans are likely to react to a scary projection of
future growth. A projection can--by itself--make things change. (In
other words, a projection may cause us to change our predictions.) Thus
one is not predicting that we will increase our production of some
chemical by a huge amount during one lifetime. One is simply projecting
that--based on past growth records--such future growth will occur
unless something changes. Sometimes the frightening implications of
growth projections are--by themselves--sufficient for people to see
that we've got to slow down some rate of growth.

[See PDF format version for Figure 1, "Typical Curve Produced by
Exponential Growth," and Table 2, "Various Powers of 2.".

--Peter Montague

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Descriptor terms: mathematics; exponential growth;