Problem 5
Question
India is currently one of the world's fastest-growing countries. By \(2040,\) the population of India will be larger than the population of China; by 2050 , nearly one-third of the world's population will live in these two countries alone. The exponential growth model \(A=574 e^{0.036 t}\) describes the population of India, \(A,\) in millions, \(t\) years after \(1974 .\) By what percentage is the population of India increasing each year?
Step-by-Step Solution
Verified Answer
The population of India is increasing each year by 3.6%.
1Step 1: Identify the rate of growth
From the given exponential growth model \(A=574 e^{0.036 t}\), the rate of growth is given by the coefficient in front of \(t\), which is 0.036.
2Step 2: Convert the rate to a percentage
To convert a decimal to a percentage, multiply by 100. So the rate of 0.036, when converted into a percentage, is \(0.036 \times 100 = 3.6\%\).
Key Concepts
Exponential FunctionPopulation GrowthMathematical Modeling
Exponential Function
In mathematics, an exponential function is a powerful tool used to describe situations where a quantity grows or decays at a rate proportional to its current value. This class of functions is easily recognized by their general form, which is usually written as...
where \( a \) is a constant that represents the initial amount, \( b \) is the base which is a positive real number, and \( x \) represents the exponent or the power.
When the base, \( b \), is greater than 1, the function models exponential growth, meaning the rate of increase accelerates over time. Conversely, if \( 0 < b < 1 \), the function models exponential decay where the quantity decreases at a rate that slows down over time. This type of function is ubiquitous in nature and technology, describing phenomena such as radioactive decay, interest compounding in finance, and indeed, population growth.
Understanding exponential functions is crucial, as they are not linear and thus do not increase by the same amount each time period but rather multiply by a certain factor. In our example, the population model of India is expressed by the equation \( A=574 e^{0.036 t} \), where \( A \) is the population in millions and \( t \) is the time in years. Here, the constant \( e \) is known as Euler's number, approximately equal to 2.71828, and it is the base of the natural logarithm, which is frequently used when dealing with continuous growth processes.
General Form of Exponential Functions
The simplest form of an exponential function can be expressed as:\( f(x) = a \times b^{x} \),where \( a \) is a constant that represents the initial amount, \( b \) is the base which is a positive real number, and \( x \) represents the exponent or the power.
When the base, \( b \), is greater than 1, the function models exponential growth, meaning the rate of increase accelerates over time. Conversely, if \( 0 < b < 1 \), the function models exponential decay where the quantity decreases at a rate that slows down over time. This type of function is ubiquitous in nature and technology, describing phenomena such as radioactive decay, interest compounding in finance, and indeed, population growth.
Understanding exponential functions is crucial, as they are not linear and thus do not increase by the same amount each time period but rather multiply by a certain factor. In our example, the population model of India is expressed by the equation \( A=574 e^{0.036 t} \), where \( A \) is the population in millions and \( t \) is the time in years. Here, the constant \( e \) is known as Euler's number, approximately equal to 2.71828, and it is the base of the natural logarithm, which is frequently used when dealing with continuous growth processes.
Population Growth
Population growth, especially within human societies, is a significant topic in both demographics and environmental studies. It refers to the change in the number of individuals in a population over time.
The model provided for India's population growth is an exponential growth model which indicates that the population size is increasing at a rate that becomes faster with each passing year. This exponential increase often cannot be sustained indefinitely due to limiting factors such as space, food, water, and other resources. On a graph, exponential growth is depicted by a J-shaped curve, which showcases the rapid escalation of population over time.
Revisiting the given formula \( A=574 e^{0.036 t} \), we can understand that the population of India is not simply increasing by the same number of individuals yearly; instead, the number of additions to the population is growing year on year. This is a typifying feature of exponential growth and has profound implications for planning in areas such as urban development, infrastructure, and resource management.
Types of Population Growth
There are two types of population growth: linear and exponential. Linear growth happens when the population increases by a constant number of individuals each period. Exponential growth, on the other hand, happens when a population increases by a constant multiplicative rate each period – necessarily leading to much quicker increases.The model provided for India's population growth is an exponential growth model which indicates that the population size is increasing at a rate that becomes faster with each passing year. This exponential increase often cannot be sustained indefinitely due to limiting factors such as space, food, water, and other resources. On a graph, exponential growth is depicted by a J-shaped curve, which showcases the rapid escalation of population over time.
Revisiting the given formula \( A=574 e^{0.036 t} \), we can understand that the population of India is not simply increasing by the same number of individuals yearly; instead, the number of additions to the population is growing year on year. This is a typifying feature of exponential growth and has profound implications for planning in areas such as urban development, infrastructure, and resource management.
Mathematical Modeling
Mathematical modeling is a methodical approach that uses mathematical language to illustrate and investigate real-world situations. By simplifying complexities and making assumptions, models enable us to predict behaviors and outcomes.
Mathematical models aren't perfect reflections of reality—instead, they serve as approximations that can significantly enhance our understanding of complex systems. In teaching mathematical modeling, it's essential to emphasize not just the construction of models but also their application and the interpretation of their results.
Steps in Mathematical Modeling
Here are the typical steps involved:- Identify the problem you wish to understand or forecast.
- Make hypotheses about the factors that affect the system and how they are related.
- Translate these relationships into mathematical equations or functions.
- Analyze the mathematical solution and interpret the results in context.
- Validate the model by comparing its predictions to real-world data.
- If necessary, adjust the model for better accuracy.
Mathematical models aren't perfect reflections of reality—instead, they serve as approximations that can significantly enhance our understanding of complex systems. In teaching mathematical modeling, it's essential to emphasize not just the construction of models but also their application and the interpretation of their results.
Other exercises in this chapter
Problem 4
Approximate each number using a calculator. Round your answer to three decimal places. \(5^{\sqrt{3}}\)
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In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions
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Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approxi
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Write each equation in its equivalent exponential form. $$5=\log _{b} 32$$
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