Problem 10 - College Algebra

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 \(1974 .\) In \(2000,\) the population of the Palestinians in the West Bank, Gaza Strip, and East Jerusalem was approximately 3.2 million and by 2050 it is projected to grow to 12 million. Use the exponential growth model \(A=A_{0} e^{k t},\) in which \(t\) is the number of years after \(2000,\) to find the exponential growth function that models the data.

Step-by-Step Solution

Answer

The exponential growth function that models the Palestinian population growth data is of the form \(A(t) = A_{0}e^{kt}\) where \(A_{0} = 3.2\) (initial population), \(t\) is the number of years after 2000 and \(k = {ln(3.75)}/{50}\) (growth rate). Plugging all these values into the function would yield the desired exponential growth function.

Step 1: Identify Starting Value

Initial given data points to the fact that the Palestinian population was about 3.2 million in the year 2000. In terms of our equation, this means that \(A_{0} = 3.2\).

Step 2: Identify Future Value

We're also given that by the year 2050, the Palestinian population is projected to grow to 12 million. Hence, \(A = 12\) for the year when \(t = 2050 - 2000 = 50\). Thus, \(A = 12\) when \(t = 50\).

Step 3: Substitute the Given Values to the Equation

Substitute \(A = 12\), \(A_{0} = 3.2\) and \(t = 50\) into the equation \(A=A_{0} e^{kt}\) to solve for \(k\). This gives us \(12 = 3.2e^{50k}\)

Step 4: Solve for the Growth Rate 'k'

Divide both sides of the equation \(12 = 3.2e^{50k}\) by 3.2 to isolate the exponential factor. We get \(e^{50k} = 12 / 3.2 = 3.75\). Taking the natural logarithm (ln) of both sides simplify the equation to \(50k = ln(3.75)\). Then, we isolate \(k\) to find its value and the final formula

Step 5: Finalizing the Exponential Growth Function

Dividing each side of the equation \(50k = ln(3.75)\) by 50 gives us the growth rate \(k\). Plugging this \(k\) value back into the general exponential growth formula \(A = A_{0}e^{kt}\) will give us the exponential growth function that models the data.

Key Concepts

Population Modeling

Population modeling is a technique used to understand how a population changes over time. In simple terms, it helps predict future population sizes based on current data. By using mathematical equations, we can create a model that reflects the growth or decline of a population.
In this specific exercise, we look at the population growth of areas like India and the Palestinian territories. With the help of exponential growth models, predictions about future populations are made. By examining data such as the current population size and expected future sizes, these models allow authorities and researchers to make important decisions concerning resource management and urban planning.
Overall, population modeling serves as a crucial tool for policy making and understanding demographic trends both locally and globally.

Exponential Functions

Exponential functions are mathematical functions of the form \(A = A_{0}e^{kt}\). They are used to describe processes where a quantity grows or decays at a constant relative rate. This makes them perfect for modeling populations, where everything grows proportionally to its current size.
In the context of our exercise, the exponential function was used to describe the growth of the Palestinian population over time. The basic idea is that each year the population grows by a percentage of its current size. This consistent growth pattern aligns perfectly with the nature of exponential functions.
Employing exponential functions allows us to predict how rapidly a population will grow or shrink over time, making it a valuable tool in many scientific and mathematical areas.

Mathematical Modeling

Mathematical modeling is the process of using mathematics to represent, analyze, and predict real-world situations. In the exercise provided, we use mathematical modeling to understand population growth. By creating a mathematical model, one can take complex situations and simplify them into understandable equations.
When modeling the Palestinian population, the given data points helped set parameters such as initial population size and expected future numbers. We translated this information into a mathematical equation using the exponential growth model. Through these calculations, the model helps us visualize and predict growth trends.
Mathematical modeling is indispensable in fields like ecology, economics, and engineering, as it allows us to simulate real-world scenarios and assess outcomes before they occur.

Growth Rate Calculation

Determining the growth rate is a fundamental aspect of using exponential models to predict future trends. The growth rate \(k\) represents the rate at which the population increases year over year. Calculating \(k\) involves examining known data points and utilizing logarithmic operations.
In our exercise, we started with known population sizes at two points in time. By substituting these values into our exponential function formula, we isolated and solved for \(k\) using natural logarithms. This calculation provided the necessary growth rate to apply in further predictions.
Understanding how to calculate growth rates enables us to assess the dynamics of population changes and is vital in various applications, including environmental studies, market analysis, and urban planning. It provides insights into how quickly changes can occur and supports long-term strategizing.