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

Town (i) has the largest growth rate of 12%. Town (ii) has the largest initial population of 1,000. Town (iv) is decreasing in size.

Step 1: Extract Percent Growth Rates

The population function is given in the form \(P=a(1+r)^t\). Here, \(r\) is the growth rate per period. For each town, we identify \(r\).(i) \( P=600(1.12)^t \): The growth rate \( r = 0.12 \), or 12%.(ii) \( P=1,000(1.03)^t \): The growth rate \( r = 0.03 \), or 3%.(iii) \( P=200(1.08)^t \): The growth rate \( r = 0.08 \), or 8%.(iv) \( P=900(0.90)^t \): The growth rate \( r = -0.10 \), or -10% (indicating a decrease).

Step 2: Determine Maximum Growth Rate

From the growth rates extracted, compare 12%, 3%, 8%, and -10%. The largest growth rate is 12% for town (i).

Step 3: Extract Initial Populations

The initial population is the value of \( a \) in the form \( P=a(1+r)^t \).(i) \( P=600(1.12)^t \): Initial population = 600.(ii) \( P=1,000(1.03)^t \): Initial population = 1,000.(iii) \( P=200(1.08)^t \): Initial population = 200.(iv) \( P=900(0.90)^t \): Initial population = 900.

Step 4: Determine Largest Initial Population

Comparing the initial populations, 1,000 is the largest from town (ii).

Step 5: Identify Decreasing Populations

A decreasing population has a growth factor less than 1. For town (iv), \(1+r = 0.90\), which is less than 1, indicating a decrease. Therefore, town (iv) is decreasing in size.

Key Concepts

Population Growth Rate

The population growth rate is a crucial concept in understanding how populations increase or decrease over time. It is usually expressed as a percentage that tells you how fast a population is changing each year. In population functions like the ones given in the exercise, the formula is typically written as \(P = a(1 + r)^t\), where \(r\) represents the growth rate per period.

In these functions:

• Town (i): \(P = 600(1.12)^t\) has a growth rate of 12%, meaning each year, the population grows by 12% from its previous size.
• Town (ii): \(P = 1,000(1.03)^t\) has a growth rate of 3%, indicating slower growth.
• Town (iii): \(P = 200(1.08)^t\) grows at a rate of 8% yearly.
• Town (iv): \(P = 900(0.90)^t\) actually shows a decrease with a rate of -10%, as the growth factor is less than 1.

The largest percent growth rate in this case is for Town (i) with 12%. This high percentage suggests a rapidly increasing population.

Initial Population

The initial population is the starting number of individuals in a population before any growth or declines take place. In mathematical terms, this is represented by \(a\) in the exponential growth formula \(P = a(1 + r)^t\). This value tells us the size of the population at time \(t = 0\).

In the problem provided:

• Town (i): \(P = 600(1.12)^t\) starts with an initial population of 600.
• Town (ii): \(P = 1,000(1.03)^t\) starts with 1,000.
• Town (iii): \(P = 200(1.08)^t\) begins with 200.
• Town (iv): \(P = 900(0.90)^t\) has an initial population of 900.

The town with the largest initial population is Town (ii), with an initial population of 1,000. This indicates the larger baseline size before any growth or decline occurs.

Decreasing Populations

Decreasing populations occur when the population number reduces over time rather than increases. This happens when the growth factor \((1 + r)\) is less than 1, making \(r\) a negative number. This indicates a negative growth rate or decline.

In the exercise:

• Town (iv): With the function \(P = 900(0.90)^t\), the growth factor is 0.90, which means the growth rate \(r\) is -10%.

Since the growth factor is less than 1, Town (iv) is experiencing a population decrease each year. This means the population is reducing by 10% annually, a sign of potentially serious demographic challenges. Understanding which populations are decreasing is essential for policy-making and planning, as it might require strategic actions to stabilize or reverse the decline.