Problem 34 - Calculus Early Transcendentals

Question

A has a current population of 500,000 people and grows at a rate of \(3 \% /\) yr. City \(B\) has a current population of 300,000 and grows at a rate of \(5 \% / \mathrm{yr} .\) a. When will the cities have the same population? b. Suppose City C has a current population of \(y_{0}<500,000\) and a growth rate of \(p>3 \% /\) yr. What is the relationship between \(y_{0}\) and \(p\) such that the Cities \(A\) and \(C\) have the same population in 10 years?

Step-by-Step Solution

Answer

(Part a) Answer: After approximately 18.73 years.

Step 1: Exponential growth formula

For each city, we can find the population at any given time using the exponential growth formula: \(P(t) = P_0 (1 + \frac{r}{100})^t\) where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time in years.

Step 2: Set up equations for cities A and B

Using the exponential growth formula, we can find an expression for the population of City A and City B at time \(t\): City A: \(P_A(t) = 500,000(1 + \frac{3}{100})^t\) City B: \(P_B(t) = 300,000(1 + \frac{5}{100})^t\)

Step 3: Find when the populations will be equal (Part a)

To find the time when the populations of City A and City B will be equal, we set the two equations equal to each other: \(500,000(1 + \frac{3}{100})^t = 300,000(1 + \frac{5}{100})^t\) Divide both sides by 100,000 to simplify: \(5(1.03)^t = 3(1.05)^t\) Now, we need to solve for \(t\). This can be done using a numerical method, such as iteration or using a graphing calculator.

Step 4: Solve for time t (Part a)

Using a graphing calculator or a computer algebra system, we find the approximate value for \(t\): \(t \approx 18.73\) So, after approximately 18.73 years, the populations of City A and City B will be the same.

Step 5: Set up equation for City C (Part b)

Suppose City C has a current population \(y_0 < 500,000\) and a growth rate of \(p > 3 \% / yr\). We want to find the relationship such that City A and City C have the same population in 10 years. Using the exponential growth formula for City A and City C, and setting the time \(t\) to 10 years: City A: \(P_A(10) = 500,000(1 + \frac{3}{100})^{10}\) City C: \(P_C(10) = y_0(1 + \frac{p}{100})^{10}\)

Step 6: Find the relationship between \(y_0\) and \(p\) (Part b)

Now, we want to find the relationship between \(y_0\) and \(p\) such that \(P_A(10) = P_C(10)\). Setting the two equations equal to each other: \(500,000(1.03)^{10} = y_0(1 + \frac{p}{100})^{10}\) Now, we solve for \(y_0\) to get the relationship between \(y_0\) and \(p\): \(y_0 = \frac{500,000(1.03)^{10}}{(1 + \frac{p}{100})^{10}}\) This equation gives the relationship between the initial population of City C, \(y_0\), and its growth rate, \(p\), such that City A and City C have the same population in 10 years.

Key Concepts

Population Growth

Population growth refers to how the number of individuals in a population increases or decreases over time. This is an essential concept in demographics and ecology, as it helps predict future population sizes, plan for resources, and understand environmental impacts.
The growth of a city can be influenced by various factors, such as birth rates, migration, and economic opportunities. In the given problem, City A and City B have different current populations and growth rates. City A's current population is 500,000 with a growth rate of 3% per year, while City B's population is 300,000, growing at 5% per year. It's crucial to understand these dynamics because they affect resource allocation and urban planning.
In numerical terms, population growth can be depicted using functions like exponential growth models, which are mathematical representations used to predict future sizes based on current data. Understanding population growth helps in making accurate predictions and informed decisions for future planning.

Exponential Function

An exponential function is a mathematical expression used to describe quantities that grow at a constant relative rate. In the context of population growth, the exponential growth formula is used to model how populations increase over time.
The formula \[ P(t) = P_0 (1 + \frac{r}{100})^t \]represents this type of growth, where:

• \(P(t)\) is the population at time \(t\)
• \(P_0\) is the initial population
• \(r\) is the growth rate
• \(t\) is the time in years

This model is appropriate for populations that grow exponentially, meaning the change in population size is proportional to the current size. As time progresses, exponential growth can lead to substantial increases, highlighting the importance of understanding the implications of growth rates.
For City A and B, substituting their values in the exponential function helps to predict at what point in time their populations will be equal. By setting their growth functions equal, we find that it takes approximately 18.73 years for the two cities to reach the same population size.

Numerical Methods

Numerical methods are mathematical techniques used to find approximate solutions to complex equations that cannot be solved analytically. In population studies, especially when dealing with exponential growth, numerical methods can provide solutions for determining timeframes or future population sizes.
In the exercise regarding Cities A and B, solving the equation \[ 5(1.03)^t = 3(1.05)^t \] is not straightforward using algebra alone. A numerical method, such as iteration, graphing calculators, or software tools, is employed to approximate the solution. These techniques involve trial and error or graphical representation to estimate the value of \(t\) when the populations will be equal.
By using these methods, students learn how to address problems involving exponential populations and other real-world mathematical models. Numerical methods thus become crucial for solving equations where traditional symbolic manipulation isn't feasible, especially in predicting and planning for future scenarios in population growth.