Problem 8 - Algebra and Trigonometry

Question

About the size of New Jersey, Israel has seen its population soar to more than 6 million since it was established. The graphs show that by \(2050,\) Palestinians in the West Bank, Gaza Strip, and East Jerusalem will outnumber Israelis. Exercises \(7-8\) involve the projected growth of these two populations. (Graph can't copy) a. 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. b. In which year will the Palestinian population be 9 million?

Step-by-Step Solution

Answer

A = 3.2 e^{kt} is the exponential growth model and the Palestinian population will be 9 million in the year calculated in step 3.

Step 1: Find the growth rate

First, we can use provided data and the formula \(A=A_{0} e^{k t}\) to find \(k\), the growth rate. Given are the following variables: \(A = 12 \text{ million}\), \(A_0 = 3.2 \text{ million}\) and \(t = 2050 - 2000 = 50\) years. By substituting the given variables into the formula and solving it for \(k\), we get \(k = \frac{1}{t} \ln(\frac{A}{A_{0}})\). Substitute the values to get \(k\).

Step 2: Formulate the Exponential Growth Function

The growth rate obtained from Step 1 is then substituted back into the formula. This determines our exponential growth function.

Step 3: Determine the Year For a Target Population

Now we use the target population to find the year in which the population will reach 9 million. Let \(A = 9\). Substitute \(A\) and \(k\) values into the formula from Step 2 and solve for \(t\). Finally, add 2000 to \(t\) to find the actual year in future.

Key Concepts

Population Growth

Population growth refers to the increase in the number of individuals within a population over time. It is an essential aspect of understanding how communities and regions change and develop.
There are several factors that contribute to population growth:

• Birth rates: The number of live births per thousand people in a year significantly influences population growth.
• Death rates: A lower death rate means more people are surviving to build up the population.
• Migration: The movement of individuals into (immigration) or out of (emigration) a region also impacts growth.

Across the globe, different areas experience population growth at varying rates due to these factors.
Understanding population growth can help planners and policymakers make informed decisions regarding resource allocation, infrastructure development, and social services.

Mathematical Modeling

Mathematical modeling is a process where mathematical structures, such as equations, represent real-world scenarios. In this context, it allows for the prediction and analysis of population growth using mathematical formulas.
With population growth, mathematical modeling often involves exponential growth models. These models are ideal when populations grow steadily over time.
An exponential growth model uses the equation:\[ A = A_0 e^{kt} \]

• A: Final population size
• A_0: Initial population size
• k: Growth rate
• t: Time in years

These models are crucial to estimating future population sizes and understanding how quickly populations may grow under certain conditions.

Growth Rate Calculation

Growth rate calculation is essential for determining how rapidly a population increases. By identifying the growth rate, we can predict future population sizes.
To find the growth rate (\( k \)), you can rearrange the exponential growth equation to:\[ k = \frac{1}{t} \ln\left(\frac{A}{A_0}\right) \]This formula uses logarithms to express the growth rate.Let's break down each component:

• \(A\): Final population we aim to reach.
• \(A_0\): Starting population.
• \(t\): Time period over which growth occurs.

By calculating this growth rate, we can define the exponential growth function, predicting how the population will shift in the coming years. Having an accurate growth rate is crucial for future planning and resource management.

Projected Population

The projected population is the estimation of future population size based on current data and calculated growth rates. Using the exponential growth model, we can predict how populations will change over time.
To calculate the projected population, plug the known values into our exponential model formula and consider the time frame you are interested in.

• Determine the target population (e.g., 9 million in the context of our exercise).
• Use the growth rate and the initial population to solve for the time it will take to reach this population size.
• Add the initial year (the year you started from, such as 2000) to the calculated time to find the future year.

This projection is key for policymakers and city planners, as it helps to prepare for future demands on resources, infrastructure, and services. Understanding and accurately predicting population changes can help address future challenges and opportunities.