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Chapter 5: Problem 22

In 2000 the Philippines and Germany had similar size populations, but very different growth rates. The population of the Philippines was 80.3 million, with a relative growth rate of \(2.0 \%\) /year. The population of Germany was 82.1 million, with a relative "growth" rate of \(-0.1 \% /\) year. Using exponential models, make a projection for the population of Germany in the year when the Philippine population has doubled.

Short Answer

Expert verified
Germany's population in 2035 will be approximately 79.2 million.

Step by step solution

01

Determine the doubling time for the Philippines

To find the year when the Philippine population will double, we use the rule of 70. The doubling time in years is calculated using the formula: \( T = \frac{70}{r} \), where \( r \) is the growth rate. For the Philippines, \( r = 2.0 \). Thus, \( T = \frac{70}{2} = 35 \) years. So, the population will double in 2035 (2000 + 35).
02

Model the population of the Philippines

We use the exponential growth model: \( P(t) = P_0 \cdot e^{rt} \). Here, \( P_0 = 80.3 \, \text{million} \), \( r = 0.02 \), and \( t = 35 \) years. This verifies our doubling: \( P(35) = 80.3 \cdot e^{0.02 \times 35} \approx 160.6 \, \text{million} \).
03

Model the population of Germany

Use the exponential model \( P(t) = P_0 \cdot e^{rt} \). For Germany, \( P_0 = 82.1 \, \text{million} \), \( r = -0.001 \), and \( t = 35 \) years. Compute: \( P(35) = 82.1 \cdot e^{-0.001 \times 35} \approx 79.2 \, \text{million} \).
04

Calculate Germany's population in 2035

Using the previous result from Step 3, we conclude that the population of Germany in 2035 will be approximately 79.2 million.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Doubling Time
Doubling time is a simple way to predict how long it will take for a population or any growing quantity to become twice its size. The concept is widely applied in exponential growth scenarios. To make this calculation easy and reliable, we use the "Rule of 70." This rule gives a straightforward formula:
  • Doubling Time (T) = \( \frac{70}{r} \)
where \( r \) is the growth rate expressed as a percentage.
For instance, if a population grows at a rate of 2% per year, like the Philippines in 2000, the doubling time can be computed by dividing 70 by 2. This results in a doubling time of 35 years, meaning the population will double by 2035.
The "Rule of 70" is a quick and rough estimate but works well for small growth rates. It helps students and analysts get a sense of how quickly growth can occur without needing to do complex calculations.
Population Projection
Population projection involves using mathematical models to estimate the future population size. This is done by applying calculated growth rates over a specified time period.
In the case of the Philippines and Germany, exponential models are used to project their future populations. The model takes into account the initial population size and the growth rate. For the Philippines, with an initial population of 80.3 million and a growth rate of 2%, the projected population in 35 years (2035) is approximately 160.6 million.
For Germany, the initial population was 82.1 million. However, with a slight decline due to a negative growth rate of -0.1%, the projection shows a decrease to about 79.2 million by 2035. These calculations help in understanding demographic changes and necessary resource planning over the years.
Exponential Models
Exponential models are mathematical representations used to describe growth or decay processes. They assume that the growth rate is proportional to the current value, producing a multiplicative effect over time. This is particularly useful for projecting populations, investments, or any phenomenon exhibiting exponential growth or decay.
  • The general formula is: \( P(t) = P_0 \cdot e^{rt} \)
where:
  • \( P(t) \) is the future value
  • \( P_0 \) is the initial value
  • \( r \) is the growth (or decay) rate
  • \( t \) is the time in years
For the Philippines, \( P_0 = 80.3 \) million and \( r = 0.02 \), yielding a doubled population in 2035. Similarly, for Germany with \( P_0 = 82.1 \) million and \( r = -0.001 \), the population is projected to slightly decrease by the same year.
Understanding exponential models is crucial for interpreting phenomena that increase or decrease at rates relative to their size. This knowledge is applied in economics, science, and resource management.

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Most popular questions from this chapter

Solve the inequalities. Where appropriate, give an exact answer as well as a decimal approximation. $$\log _{2} x \geq 0$$

The following extract is from an article by Kim Murphy that appeared in the Los Angeles Times on September 14 1994 CAIRO-Over a chorus of reservations from Latin America and Islamic countries still troubled about abortion and family issues, nearly 180 nations adopted a wide-ranging plan Tuesday on global population, the first in history to obtain partial endorsement from the Vatican. The plan, approved on the final day of the U.N. population conference here, for the first time tries to limit the growth of the world's population by preventing it from exceeding 7.2 billion people over the next two decades. (a) In 1995 the world population was 5.7 billion, with a relative growth rate of \(1.6 \% /\) year. Assuming continued exponential growth at this rate, make a projection for the world population in the year \(2020 .\) Round off the answer to one decimal place. How does your answer compare to the target value of 7.2 billion mentioned in the article? (b) As in part (a), assume that in 1995 the world population was 5.7 billion. Determine a value for the growth constant \(k\) so that exponential growth throughout the years \(1995-2020\) leads to a world population of 7.2 billion in the year 2020 .

Exercises \(55-60\) introduce a model for population growth that takes into account limitations on food and the environment. This is the logistic growth model, named and studied by the nineteenth century Belgian mathematician and sociologist Pierre Verhulst. (The word "logistic" has Latin and Greek origins meaning "calculation" and "skilled in calculation," respectively. However, that is not why Verhulst named the curve as he did. See Exercise 56 for more about this.) In the logistic model that we "I study, the initial population growth resembles exponential growth. But then, at some point owing perhaps to food or space limitations, the growth slows down and eventually levels off, and the population approaches an equilibrium level. The basic equation that we'll use for logis- tic growth is where \(\mathcal{N}\) is the population at time \(t, P\) is the equilibrium population (or the upper limit for population), and a and b are positive constants. $$\mathcal{N}=\frac{P}{1+a e^{-b t}}$$ (Continuation of Exercise 55 ) The author's ideas for this exercise are based on Professor Bonnie Shulman's article "Math-Alive! Using Original Sources to Teach Mathematics in Social Context," Primus, vol. VIII (March \(1998)\) (a) The function \(\mathcal{N}\) in Exercise 55 expresses population as a function of time. But as pointed out by Professor Shulman, in Verhulst's original work it was the other way around; he expressed time as a function of population. In terms of our notation, we would say that he was studying the function \(\mathcal{N}^{-1}\). Given \(\mathcal{N}(t)=4 /\left(1+8 e^{-t}\right)\) find \(\mathcal{N}^{-1}(t)\) (b) Use a graphing utility to draw the graphs of \(\mathcal{N}, \mathcal{N}^{-1}\), and the line \(y=x\) in the viewing rectangle [-3,8,2] by \([-3,8,2] .\) Use true portions. (Why?) (c) In the viewing rectangle [0,5,1] by \([-3,2,1],\) draw the graphs of \(y=\mathcal{N}^{-1}(t)\) and \(y=\ln t .\) Note that the two graphs have the same general shape and characteristics. In other words, Verhulst's logistic function (our \(\mathcal{N}^{-1}\) ) appears log-like, or logistique, as Verhulst actually named it in French. (For details, both historical and mathematical, see the paper by Professor Shulman cited above.)

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