Chapter 8: Exponential Astonishment Lecture notes Math 1030 Section BSection B.1: Doubling TimeDefinition of doubling timeThe time required for each doubling in exponential growth is called the doubling time. After a time t, anexponentially growing quantity with a doubling time Tdoubleincreases in size by a factor of2tTdoubleThe new value of the growing quantity is related to its initial value (at t = 0) bynew value = initial value × 2tTdoubleEx.1Consider an initial population of 10, 000 that grows with a doubling time of 10 years. What is the popu-lation after 10, 20, 30, 50 years?• In 10 years the population increases by a factor of 2:• In 20 years the population increases by a factor of 4 = 22:• In 30 years the population increases by a factor of 8 = 23:• In 50 years:1Chapter 8: Exponential Astonishment Lecture notes Math 1030 Section BEx.2Compound interest is a form of exponential growth because an interest bearing account grows by thesame percentage each year. Suppose your bank account has a doubling time of 13 years, by what factordoes your balance increase in 50 years?Ex.3 World population growth.World population doubled from 3 billion in 1960 to 6 billion in 2000. Suppose that world populationcontinues to grow with a doubling time of 40 years. What will the population be in 2030? in 2200?2Chapter 8: Exponential Astonishment Lecture notes Math 1030 Section BSection B.2: The Approximate Doubling Time FormulaDefinition of approximate doubling time formulaFor a quantity growing exponentially at a rate of P % per time period, the doubling time is approximatelyTdouble∼70PThis approximation works best for small growth rates and breaks down for growth rates over about 15%and it is called rule of 70.Ex.4World population was about 6.0 billion in 2000 and was growing at a rate of about 1.4% per year. What isthe approximate doubling time at this growth rate? If this growth rate continues, what will the populationbe in 2030?Ex.5World population doubled in the 40 years from 1960 to 2000. What was the average percentage growthrate during this period? Contrast this growth rate with the 2000 growth rate of 1.4% per year.3Chapter 8: Exponential Astonishment Lecture notes Math 1030 Section BSection B.3: Exponential Decay and Half-LifeExponential decay and half-lifeExponential decay occurs whenever a quantity decreases by the same percentage in every fixed time period.In that case, the value of the quantity repeatedly decreases to half its value, with each halving occurring ina time called the half-life = Thalf −lif e.After a time t, an exponentially decaying quantity with a half-time time Thalf −lif edecreases in size by afactor of12tThalf −lif eThe new value of the decaying quantity is related to its initial value (at t = 0) bynew value = initial value ×12tThalf −lif eEx.6You may have heard half-lives described for radioactive materials such as uranium or plutonium. Forexample, radioactive plutonium-239 (Pu-239) has a half-life of about 24, 000 years. Suppose that 100pounds of Pu-239 is deposited at a nuclear waste site. How much plutonium we have after 24, 000 years?And after 48, 000 years? And after 72, 000 years?4Chapter 8: Exponential Astonishment Lecture notes Math 1030 Section BEx.7 Carbon-14 decay.Radioactive carbon-14 has a half-life of about 5700 years. It collects in organisms only while they are alive.Once they are dead, it only decays. What fraction of the carbon-14 in an animal bone still remains 1000years after the animal has died?Ex.8 Plutonium after 100, 000 years.Suppose that 100 pounds of Pu-239 is deposited at a nuclear waste site. How much of it will still bepresent in 100, 000 years?5Chapter 8: Exponential Astonishment Lecture notes Math 1030 Section BSection B.4: The Approximate Half-Life FormulaDefinition of approximate half-life formulaThe approximate doubling time formula (the rule of 70) found earlier works equally well for exponentialdecay if we replace the doubling time with the half-life and the percentage growth rate with the percentagedecay rate. For a quantity decaying exponentially at a rate of P % per time period, the half-life is approxi-matelyThalf −lif e∼70PThis approximation works best for small decay rates and breaks down for decay rates over about 15%.Ex.9Suppose that inflation causes the value of the Russian ruble to fall at a rate of 12% per year (relative to thedollar). Approximately how long does it take for the ruble to lose half its value?6Chapter 8: Exponential Astonishment Lecture notes Math 1030 Section BSection B.5: Exact Formulas for Doubling Time and Half-LifeExact formulas for doubling time and half-lifeThe approximate doubling time and half-life formulas are useful because they are easy to remember. How-ever, for more precise work or for cases of larger rates where the approximate formulas break down, thereare exact formulas.We define the fraction growth rate as r =P100, with r positive for growth and negative for decay. For example,if the percentage growth rate is 5% per year, the fractional growth rate is r = 0.05 per year. For a 5% decayrate per year, the fractional growth rate is r = −0.05 per year.For an exponentially growing quantity with a fractional growth rate r, the doubling time isTdouble=log10(2)log10(1 + r)For an exponentially decaying quantity with a fractional growth rate r, the half-life isThalf −lif e= −log10(2)log10(1 + r)Note that the units of time used for T and r must be the same.Ex.10 Large growth rate.A population of rats is growing at a rate of 80% per month. Find the exact doubling time for this growthrate and compare it to the doubling time with the approximate doubling time formula.7Chapter 8: Exponential Astonishment Lecture notes Math 1030 Section BEx.11Suppose the Russian ruble is falling in value against the dollar at 12% per year. Using the exact half-lifeformula, determine how long it takes the ruble to lose half its