Problem 1 - Applied Calculus

Question

The following functions give the populations of four towns with time \(t\) in years. (i) \(\quad P=600(1.12)^{t}\) (ii) \(\quad P=1,000(1.03)^{t}\) (iii) \(\quad P=200(1.08)^{t}\) (iv) \(\quad P=900(0.90)^{t}\) (a) Which town has the largest percent growth rate? What is the percent growth rate? (b) Which town has the largest initial population? What is that initial population? (c) Are any of the towns decreasing in size? If so, which one(s)?

Step-by-Step Solution

Answer

(a) Town (i) has the largest growth rate at 12%. (b) Town (ii) has the largest initial population of 1000. (c) Town (iv) is decreasing in size with a growth rate of -10%.

Step 1: Understanding the Percent Growth Rate

The formula for the population of the towns is given as \( P = P_0(1 + r)^t \), where \( P_0 \) is the initial population and \( r \) is the growth rate. The percent growth rate can be determined by identifying the coefficient of \( t \) in \( (1 + r) \).

Step 2: Identify Growth Rate for Town (i)

For town (i), \( P = 600(1.12)^t \), the base is 1.12, which means \( r = 0.12 \) or 12%.

Step 3: Identify Growth Rate for Town (ii)

For town (ii), \( P = 1000(1.03)^t \), the base is 1.03, which means \( r = 0.03 \) or 3%.

Step 4: Identify Growth Rate for Town (iii)

For town (iii), \( P = 200(1.08)^t \), the base is 1.08, which means \( r = 0.08 \) or 8%.

Step 5: Identify Growth Rate for Town (iv)

For town (iv), \( P = 900(0.90)^t \), the base is 0.90, implying a decrease of \(-0.10\) or a growth rate of -10%.

Step 6: Find Town with Largest Growth Rate

Comparing the growth rates (12%, 3%, 8%, and -10%), town (i) with 12% has the largest percent growth rate.

Step 7: Identify Initial Populations

The initial population \( P_0 \) is the coefficient of the exponential term in each function: - Town (i): 600- Town (ii): 1000- Town (iii): 200- Town (iv): 900

Step 8: Find Town with Largest Initial Population

Town (ii) has the largest initial population with 1000.

Step 9: Identify Decreasing Populations

A decreasing population has a growth rate less than 0. From the analysis, town (iv) is decreasing with a growth rate of -10%.

Key Concepts

Exponential Growth

Exponential growth is a fascinating concept used to describe situations where a quantity, such as a population, increases rapidly over time. The exponential growth model in mathematics is represented by the equation: \[ P = P_0(1 + r)^t \]Here, - \(P\) stands for the population at any given time, - \(P_0\) is the initial population,- \(r\) is the growth rate per time period, - \(t\) is the time in years.
The key to understanding exponential growth is recognizing that the quantity grows by a consistent percentage rate over each time period. This means that the larger the population becomes, the faster it expands. This is evident in the growth models of towns like Town (i), Town (ii), and Town (iii) from the exercise. Each town's population increases based on their specific growth rate. Exponential growth showcases how populations can scale dramatically over time, which makes understanding its mechanisms essential.

Growth Rate Calculation

Identifying the growth rate is crucial to determining how quickly a population is changing over time. For each town in the exercise, the equation \[ P = P_0(1 + r)^t \] can be used to find the growth rate \(r\).
To calculate the percent growth rate, inspect the base of the exponential term, which appears in the form \((1 + r)\). The number right next to 1 represents the growth rate. Let's break it down:

• Town (i): The base is \(1.12\). This implies \(r = 0.12\) or a growth rate of 12%.
• Town (ii): The base is \(1.03\). Here, \(r = 0.03\) or a 3% growth rate.
• Town (iii): The base is \(1.08\). Therefore, \(r = 0.08\) or an 8% growth rate.
• Town (iv): The base is \(0.90\). This indicates a negative growth of \(-0.10\) or -10%.

Comparing these rates, Town (i) experiences the most robust growth at 12% per annum. Understanding how to calculate and interpret these rates can greatly help in analyzing population dynamics.

Initial Population

The initial population at the start of the time frame is fundamental in exponential growth models, representing the starting point for population dynamics. From the given equations, the initial population \(P_0\) is simply the constant before the exponential term, which is clear from the model \[ P = P_0(1 + r)^t \].
For each town:

• Town (i): Initial population is 600.
• Town (ii): Initial population is 1000.
• Town (iii): Initial population is 200.
• Town (iv): Initial population is 900.

Identifying the initial population value is straightforward and important in predicting future population growth. Among the analyzed towns, Town (ii) has the largest starting population with 1000, laying a foundation for significant future growth depending on their growth rate.

Decreasing Population

While exponential growth usually describes increasing populations, it can also model decreasing populations. An essential aspect of this is recognizing when populations decline due to a negative growth rate.
The equation \[ P = P_0(1 + r)^t \] indicates that a growth rate \(r\) less than zero results in a decrease. In Town (iv), the function \[ P = 900(0.90)^t \] suggests a base less than 1, specifically 0.90. This results in \(r = -0.10\), or a decrease of 10% per year.
This negative adjustment leads to a shrinking population over time. Unlike other towns, Town (iv)'s population diminishes annually because the growth factor is below 1. Recognizing a decreasing population model is instrumental in forecasting and implementing strategies that address declines.