Problem 7 - 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 Israel was approximately 6.04 million and by 2050 it is projected to grow to 10 million. Use the exponential growth model \(A=A_{0} e^{k t},\) in which \(t\) is the number of years after \(2000,\) to find an exponential growth function that models the data. b. In which year will Israel's population be 9 million?

Step-by-Step Solution

Answer

The growth constant \(k\) for the population of Israel is approximately \(ln(10/6.04)/50\) per year. Using this model, Israel's population will reach 9 million in the year \(2000 + 50*ln(9/6.04)/ln(10/6.04)\).

Step 1: Find the growth constant k.

We know that the population in 2000 (\(A_{0}\)) is 6.04 million and it's projected to grow to 10 million by 2050 (\(A\)). This means \(t = 2050 - 2000 = 50\). Substituting these values into the equation \(A=A_{0}e^{kt}\), we get \(10 = 6.04e^{50k}\). To solve for \(k\), first isolate \(e^{50k}\) by dividing both sides by 6.04, so \(e^{50k} = 10/6.04\). Then, take the natural logarithm (ln) on both sides, giving \(50k = ln(10/6.04)\). Finally, solve for \(k\) by dividing by 50, so \(k = ln(10/6.04)/50\).

Step 2: Find the year when the population reaches 9 million.

Plug in the value of \(k\) into the exponential function, making our particular model \(A = 6.04e^{(ln(10/6.04)/50) t}\). Set \(A = 9\) and solve for \(t\). This gives us the equation \(9 = 6.04e^{(ln(10/6.04)/50) t}\). First, isolate \(e^{(ln(10/6.04)/50) t}\) by dividing both sides by 6.04, resulting in \(e^{(ln(10/6.04)/50) t} = 9/6.04\). Next, take the natural logarithm on both sides to isolate \(t\), yielding \((ln(10/6.04)/50) t = ln(9/6.04)\). Finally, solve for \(t\) by multiplying both sides by 50 and dividing by ln(10/6.04). Hence, \(t = 50*ln(9/6.04)/ln(10/6.04)\). As \(t\) is the number of years after 2000, add 2000 to \(t\) to find the actual year.

Key Concepts

Exponential Functions

Exponential functions are a crucial concept in mathematics, particularly for modeling growth processes like population increase. They can describe quantities that grow at a consistent relative rate. This is done using the formula: \[ A = A_0 e^{kt} \] where \( A_0 \) is the initial amount, \( k \) is the growth constant, and \( t \) is the time elapsed.

• \( A \) represents the amount at time \( t \).
• \( e \) is the base of the natural logarithm, approximately 2.71828.
• \( k \) indicates whether the process describes growth (if positive) or decay (if negative).

Exponential functions are essential because they appear naturally in real-life scenarios where the rate of growth is proportional to the current amount. Understanding how to manipulate and use these functions provides important insights into phenomena such as population growth, investments, and radioactive decay.

Population Growth

Population growth is when the number of individuals in a population increases over time. Human populations often follow an exponential growth pattern under favorable conditions. This means they grow faster as they become larger. The formula for this growth is \( A = A_0 e^{kt} \). In real-life scenarios, factors like resources, environment, and policies may alter this growth.
When predicting future population sizes, models like these enable us to understand trends and make informed decisions for urban development and resource management. Specifically, in the case of Israel, the projected growth from 6.04 million in 2000 to 10 million in 2050 demonstrates how this model can be used to predict future scenarios.
It's essential to consider these models are ideal representations and actual growth may differ due to external factors and constraints.

Logarithms

Logarithms are mathematical tools used to solve for variables in exponential equations. They convert multiplication into addition, making them extremely helpful in solving exponential functions like population growth models. The natural logarithm, represented as \( \ln \), is especially important for solving equations involving base \( e \).
Consider the equation \( A = A_0 e^{kt} \). To isolate \( t \) or \( k \), we often use \( \ln \). For instance, to solve for \( k \), you would first divide both sides by \( A_0 \) and then take the \( \ln \) of both sides:

• \( \ln(e^{kt}) = \ln(\frac{A}{A_0}) \)
• Using the property \( \ln(e^x) = x \), we get \( kt = \ln(\frac{A}{A_0}) \)

Understanding logarithms is vital as they enable us to unravel exponential models and predict variables, much like solving for when Israel’s population reaches 9 million.

Growth Constant

The growth constant, denoted by \( k \), is a crucial component of the exponential growth model. It represents the rate at which a population grows. In the model \( A = A_0 e^{kt} \), \( k \) helps us understand how fast or slow the population is increasing over time.
To find \( k \), we use known data points and logarithmic functions, as seen in the original problem where Israel's population grows from 6.04 to 10 million from 2000 to 2050. Solving for \( k \) involves using the equation: \[ k = \frac{\ln(\frac{A}{A_0})}{t} \] This equation tells us that the growth constant is dependent on the relative increase and the time frame.

• When \( k \) is positive, it indicates growth.
• If \( k \) is zero, the population remains constant.
• Negative \( k \) suggests a declining population.

This constant is not just a number; it provides meaningful insights into how the population will change, allowing planners and analysts to make informed predictions.