RACHEL's Hazardous Waste News #197

=======================Electronic Edition========================

RACHEL’S HAZARDOUS WASTE NEWS #197
—September 5, 1990—
News and resources for environmental justice.
——
Environmental Research Foundation
P.O. Box 5036, Annapolis, MD 21403
Fax (410) 263-8944; Internet: erf@igc.apc.org
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HOW TO PROJECT FUTURE GROWTH.

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.

Increasing
Quantity
(of anything)
.
150                                    *
.     Typical Curve Produced By
.     Exponential Growth               *
125   (Growth rate = 10% per year)
.                                     *
.
100                                 *
.
.                                 *
75
.                              *
.
50                        *
.
.                   *
25
.             *
.* * *  *
0 _______________________________________
.
1940   1950   1960   1970   1980   1990
.              TIME

Figure 1. Typical curve created by something growing exponentially.
Notice that at the beginning, the curve is not rising steeply; as
time passes, however, the curve becomes steeper and steeper. The
larger the quantity becomes, the faster it grows; this is the main
characteristic of things that grow exponentially.

======================================

Table 1. Various Powers of 2

If p equals Then 2^p equals
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1024
11 2048
12 4096
13 8192
14 16,384
15 32,768
16 65,536
17 131,072
18 262,144
19 524,288
20 1,048,576

Table 1. The value of 2 raised to the power of various annual rates of
increase, p, expressed as a percentage.

–Peter Montague, Ph.D.

Descriptor terms: mathematics; predictions; exponential growth;

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