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 Td 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 = Td = 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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