# Doubling Time and Exponential Growth

## Contents

#### Doubling Time and Exponential Growth

Many phenomena (such as population growth, savings in a bank, and the spread of an epidemic) behave in a nonlinear fashion such that their changes vary in an exponential manner from one year to the next. An important characteristic of exponential growth is that a quantity grows by a certain percentage each year. This might not seem to be much at the beginning, but the cumulative amount can become quite staggering over time. For example, consider a city of 100,000 inhabitants. Because of immigration, increased birth and decreased death rates, its population expands at a rate of 5% annually. After the end of the first year, population will grow to 105,000 and then to 110,250 at the end of the second year.

A simple calculation shows that this city will double to 200,000 people in about 14 years, double again to 400,000 in 28 years, reach 3 million after the passing of one generation (70 years) and grow to over 13 million after a century. It is constructive to give a convenient method to express the growth of quantities that vary in this fashion (that is, those which grow by a fixed percentage every year) in terms of their doubling time. Doubling time is defined as the time it takes a quantity to double, and can be reasonably approximated by dividing 70 by the percentage growth rate:( 1 )

${T_2}=\frac{70}{Percentage\qquad Rate\qquad of\qquad Growth} \qquad \qquad(1)$

(1-2)

As we will see in Chapter 11, the same equation can be used to calculate the time which it takes a sample which decays in population (i.e. has a negative growth rate) to reduce in half. This is known as half-life, an example of which is the activity of radioactive material or any other sample which decreases yearly by a fixed percentage.

Example 1-3: A couple is planning to save money to pay for their new-born son’s college tuition by putting aside $1000 a year. How much would the child have when he enters college at 18-years of age, if his parents decided to deposit the money in a) a safe deposit box, or b) a saving account that paid 7% interest annually? Solution: If the money is put in a deposit box, the growth is linear—the annual increase is always the same. The saving is$1,000 after the first year, $2,000 after the second year, etc. At age 18, the child will have$18,000 in savings.

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The Incredible Power of Two
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Figure 1: Linear vs. Exponential Growths

ex 1-3

Old French Riddle: At first there is only one lily pad in the pond, but the next day it doubles, and thereafter each of its descendants doubles. The pond completely fills up with lily pads in 30 days. When is the pond half full? Answer: on the 29th day.

If the money is invested in a saving account, the percentage rate of increase is the same each year but the total increase is exponential. Let’s say the parents deposit $1,000 on the child’s first birthday. The bank pays 7% ($70) in interest for the first year. With the additional $1000 deposit at the end of the first year, the total money saved after the second birthday is 1,000(1.07) + 1,000 =$2,070, and 2,070(1.07) + 1,000 = $3,215 after the third. By his 18th birthday, the total deposit has grown to nearly$34,000 instead of \$18,000 he would have had if no interests were paid.( 2 ) The difference between the linear and exponential growths becomes substantial by the time the boy is ready for college.

Example 1-4: It is estimated that the world’s petroleum production will peak sometime around 2005-2010, at which time we will have used up one half of all our petroleum reserves. If the total cumulative amount of oil which has been produced from the early days of discovery in the 1860s until today is estimated at 800 billion barrels, how long would it take before we deplete all our petroleum reserves? According to the latest projections, Energy Information Administration(2005) the world energy consumption will increase by 57% between 2002 and 2025.

Solution: What may come immediately to mind is that since it took us 145 years to use half of our reserves (1860-2005), we will have oil for another 145 years. This is unfortunately far from the truth! In fact we will use as much oil as we have previously used in only one doubling time. Assuming demand continues to grow at the rate of 2.47% a year (57% in 23 years), the doubling time is calculated from Equation 1 as 70/2.47 = 28.3 years. Unless we cut consumption or find new reserves of oil and new sources of energy, we expect to run out of oil approximately 28 years after its peak -- by 2033-2038.

Example 1-5: According to a US Census, in 1973, the national birth and death rates were 15.6 and 9.4 per thousand. Furthermore, the Immigration and Naturalization Services (INS) data showed that 400,000 people or roughly 1.5 persons for every 1,000 people immigrated into the United States during that year. Calculate the population growth rate in 1973. If the population were to increase at the same rate, when would the US population reach twice that of 1973?

Solution: The growth per 1,000 persons was 15.6-9.4+1.5 = 7.7 or 0.77%. If growth were to continue at this rate, the US population would double the 1973 population in 70/0.77 = 91 years, in year 2064.

Example 1-6: What is the rate of population increase in a community in which there are only two children per family?

Answer: The community has zero rate of population increase. Essentially, each parent replaces their children for themselves.

Example 1-7: A bacterial colony fills up a jar at 12:00 pm. If the bacteria double in population every minute, when is the jar 1/16 full?

Answer: The jar is half full one minute before noon at 11:59, 1/4 full at 11:58, 1/8 full at 11:57, and 1/16 full at 11:56.

The data on the use of petroleum shows that the world had used only 1/8 of its reserves in 1973, and the consumption has doubled every ten years for the last few decades. As of the writing of this text, roughly half of all petroleum reserves are used up; we are at 11:59 hours. The clock ticked 11:56 am during the oil crisis of early 1970s.

Example 1-8: A popular brand of antibacterial cleaner advertises that it kills 99.9% of all kitchen bacteria. Assuming that bacteria can double their population every 24 hours, how often must the disinfectant be applied to maintain kitchen bacteria in check?

Answer: Assuming the company’s claim is accurate, after the antibacterial application 0.1% (or 1 in every 1000) bacteria remain. For the population to reach its original level, it must increase 1000 fold or about 10 doubling times. The disinfectant must be applied at least once every 10 days!

## References

“World Energy and Economic Outlook”, Energy Information Administration, DOE/EIA-0484, July 2005.

(1) Equation 1 assumes that growth is taking place continuously. This is definitely true in the case of population growth and energy consumption. In many other instances, such as interest paid by a bank, growth comes in steps. For example a bank that pays 3% interest per annum compounded monthly pays 3/12=0.25% each month which results in a slightly different growth rate. As we will see in Chapter 15, banks and other financial institutions often apply the “Rule of 72” - divide 72 by the interest rate to find the time in which a saving deposit doubles in value.

(2) The cumulative value of all payments can be calculated by applying Equation (15-4) directly.