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Chapter 1: Problem 1

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)?

Short Answer

Expert verified
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 by step solution

01

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).
02

Determine Maximum Growth Rate

From the growth rates extracted, compare 12%, 3%, 8%, and -10%. The largest growth rate is 12% for town (i).
03

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.
04

Determine Largest Initial Population

Comparing the initial populations, 1,000 is the largest from town (ii).
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Write the functions in Problems \(21-24\) in the form \(P=P_{0} a^{t}\) Which represent exponential growth and which represent exponential decay? $$P=15 e^{0.25 t}$$

A quantity \(P\) is an exponential function of time \(t .\) Use the given information about the function \(P=P_{b} a^{t}\) to: (a) Find values for the parameters \(a\) and \(P_{0}\). (b) State the initial quantity and the percent rate of growth or decay. \(P=1600\) when \(t=3\) and \(P=1000\) when \(t=1\)

Most breeding birds in the northeast US migrate elsewhere during the winter. The number of bird species in an Ohio forest preserve oscillates between a high of 28 in June and a low of 10 in December. \(^{97}\) (a) Graph the number of bird species in this preserve as a function of \(t,\) the number of months since June. Include at least three years on your graph. (b) What are the amplitude and period of this function? (c) Find a formula for the number of bird species, \(B\), as a function of the number of months, \(t\) since June.

(a) What is the continuous percent growth rate for \(P=\) \(100 e^{0.06 t},\) with time, \(t,\) in years? (b) Write this function in the form \(P=P_{0} a^{t} .\) What is the annual percent growth rate?

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