- Garden Mosaics projects promote science education while connecting young and old people as they work together in local gardens.
- Hope Meadows is a planned inter-generational community containing foster and adoptive parents, children, and senior citizens
- In August 2002, the Los Angeles Unified School District (LAUSD) Board voted to ban soft drinks from all of the district’s schools

C02 emissions / carbon dioxide
chemical hazards
chemicals and health
chemicals and health
children's health
children / youth
climate change / global warming
co2 emissions
coal industry / coal power
disease
environmental health
EPA / Environmental Protection Agency
failure of government
fish / fishing
food safety / security
global warming
greenhouse gases / greenhouse effect
landfills/solid waste
oceans
PBDEs
pesticides
precautionary principle / primary prevention
public health
toxics

Published September 4, 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;