A simple way to look out for exponential growth is to try to spot a doubling time. A concerned newspaper reader in the Spring of 2020 might notice the apparent doubling between the 23rd and 26th of February, for example, and then keep watching the news to see if cases continue to double approximately every three days.
World population growth accelerated after World War II, when the population of less developed countries began to increase dramatically. After millions of years of extremely slow growth, the human population indeed grew explosively, doubling again and again; a billion people were added between 1960 and 1975; another billion were added between 1975 and 1987. Throughout the 20th century each additional billion has been achieved in a shorter period of time. Human population entered the 20th century with 1.6 billion people and left the century with 6.1 billion.
investigation 20 doubling time exponential growth answer key.zip
A story said to have originated in Persia offers a classic example of exponential growth. It tells of a clever courtier who presented a beautiful chess set to his king and in return asked only that the king give him one grain of rice for the first square, two grains, or double the amount, for the second square, four grains (or double again) for the third, and so forth. The king, not being mathematically inclined, agreed and ordered the rice to be brought from storage. The eighth square required 128 grains, the 12th took more than one pound. Long before reaching the 64th square, every grain of rice in the kingdom had been used. Even today, the total world rice production would not be enough to meet the amount required for the final square of the chessboard. The secret to understanding the arithmetic is that the rate of growth (doubling for each square) applies to an ever-expanding amount of rice, so the number of grains added with each doubling goes up, even though the rate of growth is constant.
The number of years required for the population of an area to double its present size, given the current rate of population growth. Population doubling time is useful to demonstrate the long-term effect of a growth rate, but should not be used to project population size. Many more-developed countries have very low growth rates. But these countries are not expected to ever double again. Most, in fact, likely have population declines in their future. Many less-developed countries have high growth rates that are associated with short doubling times, but are expected to grow more slowly as birth rates are expected to continue to decline.
This is the second post in a three-part series about exponential growth and doubling time. This post will explore the concept of doubling time and explain how one can calculate the doubling time for a population growing exponentially using the rule of 70.
Doubling time is the amount of time it takes for a given quantity to double in size or value at a constant growth rate. We can find the doubling time for a population undergoing exponential growth by using the Rule of 70. To do this, we divide 70 by the growth rate (r).
The visualization shows how strongly the growth rate of the world population changed over time. In the past the population grew slowly: it took nearly seven centuries for the population to double from 0.25 billion (in the early 9th century) to 0.5 billion in the middle of the 16th century. As the growth rate slowly climbed, the population doubling time fell but remained in the order of centuries into the first half of the 20th century. Things sped up considerably in the middle of the 20th century.
Since then, population growth has been slowing, and along with it the doubling time. In this visualisation we have used the UN projections to show how the doubling time is projected to change until the end of this century. By 2100, it will once again have taken approximately 100 years for the population to double to a predicted 10.8 billion.
We're releasing an analysis showing that since 2012, the amount of compute used in the largest AI training runs has been increasing exponentially with a 3.4-month doubling time (by comparison, Moore's Law had a 2-year doubling period).[1] Since 2012, this metric has grown by more than 300,000x (a 2-year doubling period would yield only a 7x increase). Improvements in compute have been a key component of AI progress, so as long as this trend continues, it's worth preparing for the implications of systems far outside today's capabilities.
Much of the growth in the world around us is exponential. The curve in Figure 1 depicts exponential growth for any of the above examples. Notice that each of the successive equal time intervals on the horizontal scale corresponds to a doubling of the quantity on the vertical scale. This doubling of quantity becomes startling when you find the car you financed to purchase costs nearly twice as much as if you paid cash. When the growth of a quantity is exponential, the amount doubles in a certain interval of time. We speak of doubling time.
The importance of the exponential curve of Figure 1 is that the time required for the growing quantity to double in size, a 100% increase, is a constant. For example, if the population of a growing city takes 10 years to double from 100,000 to 200,000 inhabitants and its growth remains exponential, then in the next 10 years the population will double to 400,000 and 10 years after that to 800,000 and so on.
Steady growth in a finite environment gets interesting. Consider bacterial growth by cell division, in which one bacterium becomes two, the two divide to become four, the four divide to become eight, and so on. Suppose the division time for a certain strain of bacteria is 1 minute. In this steady growth the number of bacteria grows exponentially with a doubling time of 1 minute. Further, suppose that one bacterium is put in a bottle at 11:00 a.m. along with adequate food, and that growth continues steadily until the bottle becomes full of bacteria at noon.
It is interesting and important to note that each square contains one more grain than all the preceding squares combined. This is true anywhere on the board. Note that when eight grains are placed on the fourth square, the eight is one more than all previous grains of wheat, the total of seven grains that were already on the board. Or the 32 grains placed on the sixth square is one more than all previous grains of wheat, a total of 31 grains that were already on the board. We see that in one doubling time we add more than all that had been added in all the preceding growth! To repeat for emphasis: In one doubling time more growth occurs than in all preceding growth combined!
Where the doubling time of virus cases in some cities can be three or four days, continued exponential growth would multiply the cases more than 250-fold in one month! The smaller the doubling time, the sooner other factors come into play to end the exponential phase. What feeds the growth eventually subsides, as with the 1918 flu epidemic.
This lesson covers doubling time, its formula, and equations. One doubling time definition is the time it takes for a population to double when growing at a constant rate. It has many applications, particularly in studying and predicting population dynamics and financial growth. The doubling time formula, eqDoubling\ time = t ln 2 / [ ln (1 + r/100) ] /eq, is used to calculate doubling time. For example, it would take a population 14 years to double at a growth rate of 5% per year, and that can be found using the doubling time formula. The formula can be simplified to the Rule of 69, Rule of 72, and Rule of 70 using assumptions and applying them to particular scenarios. These simplified equations are: eqdoubling\ time = 69 / r , doubling\ time = 72 / r , and\ doubling\ time = 70 / r /eq, respectfully. Population doubling time can be found using the doubling time formula and/or equation. These rules are used to calculate doubling time, as well as the doubling time formula. The Rule of 70 relies on the assumption that the annual growth rate will stay consistent, it calculates exponential growth, and it is the set number we use to calculate doubling time.
The Rule of 70 is a simplified way of determining the doubling time using the equation, doubling time = 70 / r , where r is the rate of growth for a population in percent. For example, if a population of 10 species were growing by two individuals a year, the r value would be 20%.
In the following problems, students will calculate the doubling time for different populations using the rule of 70. The annual rate of growth will also be solved for given the doubling time of the population.
3. Your friend calculated the doubling time for a population with annual growth rate of 5% as follows: dt = 70 / 0.05 = 1,400 years. What is wrong with this calculation? What is the correct doubling time?
Around the world, people in similar situations tend to reproduce at similar rates, and so the rates of growth throughout human history have been surprisingly consistent. So, ideas like doubling time are very useful to help us prepare for the future. What sort of infrastructure will we need? How much space will a population need? What amount of resources will they use? These questions all require an understanding of future population sizes. So, this is an important field of study. Let's get to it, on the double. 2ff7e9595c
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