# Exponential Growth and the Rule of 70 - DOC - DOC

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```					                 Exponential Growth and the Rule of 70

An important feature of positive feedback occurs with exponential growth. Growth is
exponential when it occurs at a constant rate per time period. (rather than a constant
amount) Exponential growth produces a “J” curve.

Calculating exponential growth

If we know the Growth Rate (R) the percentage of change per unit time,
we can calculate the growth constant (k) using the formula k = R/100

so for example, if R is 2% then k = +0.02

If k is positive we have exponential growth whereas,
If k is negative we have exponential decay. (Radioactive decay)

We can use the Exponential Growth Formula to calculate the future size of a population
if we know the growth rate.
kt
The Exponential Growth Formula is N = Noe

where,         N = future population size
No = current population size
k = growth rate (as a decimal)
t = time
e = ex = base of natural logarithms = 2.71828

Example calculation:

If the world human population was 6.3 billion people in 2003, how many people will
there be in 2020 if the growth rate is 1.36% per year?

N = Noekt
9               0.0136 X 17
N = 6.3 X 10 X 2.17828
9               0.2312
N = 6.3 X 10 X 2.17828
9
N = 7.94 X 10 people in 2020
The Doubling time (Td ) is the time necessary to double for the quantity being measured
to double.

If we want N to be 2No , we can calculate the doubling time with the following formula:

2No = NoekTd

where, Td is the doubling time.

Taking the natural logarithm of both sides we get

ln 2 = k T d or    Td = ln 2 / k

Remember:      k = R/100

then,          Td = 0.693/ R/100

Td = 69.3/R

From this we get the Rule of 70

Doubling time = 70 / growth rate or     Td = 70/R
Sample problem:

How many years will it take for the human population to double if the growth rate is 2%
per year?

Answer:        70/R = T d    =        70/2 = 35 years

Problem: How many years will it take for the United States to double its population size
assuming a growth rate of 1.36% and no immigration?

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