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Chapter 4: Problem 1

Population growth: The following table shows the size, in thousands, of an animal population at the start of the given year. ? Find an exponential model for the population. $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { (thousands) } \end{array} \\ \hline 2004 & 2.30 \\ \hline 2005 & 2.51 \\ \hline 2006 & 2.73 \\ \hline 2007 & 2.98 \\ \hline 2008 & 3.25 \\ \hline \end{array} $$

Short Answer

Expert verified
The exponential model for the population is \( P(t) = 2.30 \times e^{0.0916t} \).

Step by step solution

01

Understanding the Exponential Model

An exponential model can be represented by the function \( P(t) = P_0 imes e^{rt} \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population at \( t=0 \), \( r \) is the growth rate, and \( t \) is time in years.
02

Setting the Initial Conditions

The initial condition can be set from the year 2004, which corresponds to \( t=0 \) thus \( P_0 = 2.30 \). The population in 2004 is 2.30 thousand, which establishes the initial population size.
03

Finding the Growth Rate

Use the population data for 2008 (4 years after 2004) to solve for \( r \). From the table, \( P(4) = 3.25 \). Substitute into the model: \( 3.25 = 2.30 imes e^{4r} \). Solve for \( r \) by dividing both sides by 2.30 and taking the natural logarithm:\[ e^{4r} = \frac{3.25}{2.30} \quad \Rightarrow \quad 4r = \ln\left(\frac{3.25}{2.30}\right) \quad \Rightarrow \quad r = \frac{1}{4} \ln\left(\frac{3.25}{2.30}\right) \].
04

Calculating the Growth Rate

Compute \( r \) using the formula derived in Step 3: \[ r = \frac{1}{4} \ln\left(\frac{3.25}{2.30}\right) \approx \frac{1}{4} \times 0.3665 \approx 0.0916 \] which indicates a growth rate of approximately 9.16% per year.
05

Formulating the Exponential Model

With \( P_0 = 2.30 \) and \( r \approx 0.0916 \), the exponential model is \[ P(t) = 2.30 imes e^{0.0916t} \]This function describes the exponential growth of the animal population over time, starting from the year 2004.

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Key Concepts

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

Population Growth
Population growth is a fascinating and essential topic when studying biology, ecology, and even economics. When we talk about population growth, we are discussing how the size of a particular group, such as an animal species in a specific area, changes over time. This growth can be influenced by numerous factors, including birth rates, death rates, immigration, and emigration.

However, when we refer to exponential growth specifically, we focus on populations that grow in a way where the increase is proportional to the current size of the population. This means the larger the population becomes, the faster it has the potential to grow, assuming there are no limitations like food or space. Therefore, understanding and predicting how populations grow over time becomes crucial for planning and conservation.
Animal Population
When examining animal populations, it's important to understand what factors can impact their growth patterns. Take, for example, a table of population data that refers to an animal population increasing over several years. In our case, each row in the table represents the number of animals, in thousands, at the start of each year.

The concept of an animal population focuses on the collective group of a specific species. Understanding animal populations helps conservationists manage wildlife reserves and protect endangered species. By knowing how a population grows, experts can make informed decisions about measures needed to maintain a healthy ecosystem.
Growth Rate
The growth rate is a critical concept in understanding how populations change over time. In exponential growth models, the growth rate is symbolized as "r" in our mathematical equation. This rate reflects how quickly or slowly a population increases during a specific time frame.

Calculating the growth rate involves understanding the changes in population size over years and using functions such as natural logarithms when dealing with exponential growth.
For example, the growth rate in an exponential model tells us how much of a population increase we can expect each year. If the calculated growth rate is 9.16%, as was deduced from our data, it means that each year the population is expected to grow by nearly 9.16% over the previous year.
Mathematical Modeling
Mathematical modeling is a powerful tool used to represent real-world situations using mathematical concepts and structures. When it comes to population studies, models allow us to predict future changes and trends based on current data.

The exponential model used in this exercise is expressed as the function \[ P(t) = P_0 \times e^{rt} \]where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( r \) is the growth rate, and \( e \) is the base of natural logarithms (approximately equal to 2.718).
Using this model, we can plug in the known values to calculate the population at any given time \( t \). Such models are hugely valuable in ecology, as they help scientists make informed decisions about conservation and resource management.

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Most popular questions from this chapter

Decibels: Sound exerts a pressure \(P\) on the human ear. This pressure increases as the loudness of the sound increases. It is convenient to measure the loudness \(D\) in decibels and the pressure \(P\) in dynes per square centimeter. It has been found that each increase of 1 decibel in loudness causes a \(12.2 \%\) increase in pressure. Furthermore, a sound of loudness 97 decibels produces a pressure of 15 dynes per square centimeter. a. Explain why \(P\) is an exponential function of \(D\) and find the growth factor. b. Find \(P(0)\) and explain in practical terms what your answer means. c. Find an exponential model for \(P\) as a function of \(D\). d. When pressure on the ear reaches a level of about 200 dynes per square centimeter, physical damage can occur. What decibel level should be considered dangerous?

State quakes: The largest recorded earthquake centered in Idaho measured \(7.2\) on the Richter scale. a. The largest recorded earthquake centered in Montana was \(3.16\) times as powerful as the Idaho earthquake. What was the Richter scale reading for the Montana earthquake? b. The largest recorded earthquake centered in Arizona measured \(5.6\) on the Richter scale. How did the power of the Idaho quake compare with that of the Arizona quake?

Inflation: The yearly inflation rate tells the percentage by which prices increase. For example, from 1990 through 2000 the inflation rate in the United States remained stable at about \(3 \%\) each year. In 1990 an individual retired on a fixed income of \(\$ 36,000\) per year. Assuming that the inflation rate remains at \(3 \%\), determine how long it will take for the retirement income to deflate to half its 1990 value. (Note: To say that retirement income has deflated to half its 1990 value means that prices have doubled.)

Fully stocked stands: This is a continuation of Exercise 11. In this exercise we study one of the ingredients used in formulating the relation given in the preceding exercise among the stand-density index \(S D I\), the number \(N\) of trees per acre, and the diameter \(D\), in inches, of a tree of average size. \({ }^{34}\) This ingredient is an empirical relationship between \(N\) and \(D\) for fully stocked stands - that is, stands for which the tree density is in some sense optimal for the given size of the trees. This relationship, which was observed by L. H. Reineke in 1933 , is $$ \log N=-1.605 \log D+k $$ where \(k\) is a constant that depends on the species in question. a. Assume that for loblolly pines in an area the constant \(k\) is 4.1. If in a fully stocked stand the diameter of a tree of average size is 8 inches, how many trees per acre are there? (Round your answer to the nearest whole number.) b. For fully stocked stands, what effect does multiplying the average size of a tree by a factor of 2 have on the number of trees per acre? c. What is the effect on \(N\) of increasing the constant \(k\) by 1 if \(D\) remains the same?

The pH scale: Acidity of a solution is determined by the concentration \(H\) of hydrogen ions in the solution (measured in moles per liter of solution). Chemists use the negative of the logarithm of the concentration of hydrogen ions to define the \(\mathrm{pH}\) scale: $$ \mathrm{pH}=-\log H . $$ Lower pH values indicate a more acidic solution. a. Normal rain has a pH value of 5.6. Rain in the eastern United States often has a pH level of \(3.8\). How much more acidic is this than normal rain? b. If the \(\mathrm{pH}\) of water in a lake falls below a value of 5 , fish often fail to reproduce. How much more acidic is this than normal water with a \(\mathrm{pH}\) of \(5.6 ?\)
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