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Chapter 4: Problem 3

A population with given per capita growth rate: A certain population has a yearly per capita growth rate of \(2.3 \%\), and the initial value is 3 million. a. Use a formula to express the population as an exponential function. b. Express using functional notation the population after 4 years, and then calculate that value.

Short Answer

Expert verified
The population after 4 years is approximately 3,283,720.

Step by step solution

01

Understand the problem

We are given a population with an initial size of 3 million and a yearly per capita growth rate of 2.3%. We need to formulate an exponential function to model the population over time, and then find the population after 4 years.
02

Formula for exponential growth

The general formula for exponential growth is given by \[ P(t) = P_0 (1 + r)^t \]where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is time in years.
03

Plug in the given values

Given that the initial population \( P_0 \) is 3 million and the growth rate \( r \) is 2.3%, or 0.023 in decimal form, the function becomes \[ P(t) = 3,000,000 (1 + 0.023)^t \].
04

Express population after 4 years

To find the population after 4 years, substitute \( t = 4 \) in the formula: \[ P(4) = 3,000,000 (1 + 0.023)^4 \].
05

Calculate the expression

First compute the base inside the bracket: \( 1 + 0.023 = 1.023 \).Then, raise 1.023 to the power of 4: \[ 1.023^4 \approx 1.09457 \].Now multiply by the initial population: \[ P(4) \approx 3,000,000 \times 1.09457 \approx 3,283,720 \].
06

Conclude the results

Thus, using the exponential model, the population after 4 years is approximately 3,283,720.

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Key Concepts

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

Per Capita Growth Rate
In population studies, the term "per capita growth rate" refers to the growth rate of a population per individual within a given time frame. It is commonly expressed as a percentage, which helps in understanding how much the overall population increases for each individual.
For example, if a population has a per capita growth rate of 2.3%, as given in our exercise, this means that each individual in the population contributes to a 2.3% increase per year. This rate is crucial for assessing how rapidly a population is growing and can influence decision-making in areas like resource allocation and environmental planning.
  • The per capita growth rate is calculated by taking the annual increase in population divided by the initial population size.
  • It can also be denoted in decimal form (e.g., 2.3% is represented as 0.023) to be used directly in mathematical equations.
Knowing the per capita growth rate can help predict future size and growth trends by allowing us to plug this value into an exponential growth model.
Exponential Function
The exponential function is a mathematical expression that represents growth. It is characterized by a constant rate of growth relative to the current size. In our context, the formula for exponential growth is \[ P(t) = P_0 (1 + r)^t \].
  • \( P(t) \): Population at time \( t \).
  • \( P_0 \): Initial population size.
  • \( r \): Growth rate, in decimal.
  • \( t \): Time elapsed.
Because the exponential function multiplies the initial population by a growth factor per year, it models continuous and compounded growth over time.
This type of function is particularly useful because it mirrors real-world scenarios where increases happen at a relative rate rather than an absolute amount. In practical terms, this means that as a population grows, the absolute increase becomes larger each year because it is based on the continually increasing population number itself.
Population Modeling
Population modeling involves using mathematical formulas and equations, like exponential functions, to represent how populations change over time. By creating these models, we can effectively simulate and forecast population behavior under various scenarios.
In our example, we use the exponential function to model the population's growth due to its per capita rate.
  • This approach assumes that the rate stays constant, which may not always be true in a real-world context, where resources or other factors can influence growth rates.
  • Models allow for calculations of population size at any given future point, such as determining that the population will be approximately 3,283,720 after four years at the given rate.
The primary benefit of population modeling is its ability to assist in planning for future needs, such as infrastructure, education, and healthcare, based on projected growth. This quantitative framework gives researchers, policymakers, and planners the tools they need to make informed decisions.

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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?

Cell phones: The following table shows the number, in millions, of cell phone subscribers in the United States at the end of the given year. $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Subscribers (millions) } \\ \hline 2001 & 128.4 \\ \hline 2002 & 140.8 \\ \hline 2003 & 158.7 \\ \hline 2004 & 182.1 \\ \hline 2005 & 207.9 \\ \hline \end{array} $$ a. Plot the natural logarithm of the data points. Does this plot make it look reasonable to approximate the original data with an exponential function? b. Find the regression line for the natural logarithm of the data and add its graph to the plot in part a. c. Construct an exponential model for the original subscribership data using the logarithm as a link.

Headway on four-lane highways: When traffic is flowing on a highway, the headway is the average time between vehicles. On four-lane highways, the probability \(P\) that the headway is at least \(t\) seconds is given to a good degree of accuracy \({ }^{6}\) by $$ P=e^{-q t}, $$ a. On a four-lane highway carrying an average of 500 vehicles per hour in one direction, what is the probability that the headway is at least 15 seconds? (Note: 500 vehicles per hour is \(\frac{500}{3600}=0.14\) vehicle per second.) b. On a four-lane highway carrying an average of 500 vehicles per hour, what is the decay factor for the probability that headways are at least \(t\) seconds? Reminder: An important law of exponents tells us that \(a^{b c}=\left(a^{b}\right)^{c}\). where \(q\) is the average number of vehicles per second traveling one way on the highway.

Weight gain: Zoologists have studied the daily rate of gain in weight \(G\) as a function of daily milkenergy intake \(M\) during the first month of life in several ungulate (that is, hoofed mammal) species. \({ }^{31}\) (Both \(M\) and \(G\) are measured per unit of mean body weight.) They developed the model $$ G=0.067+0.052 \log M, $$ with appropriate units for \(M\) and \(G\). a. Draw a graph of \(G\) versus \(M\). Include values of \(M\) up to \(0.4\) unit. b. If the daily milk-energy intake \(M\) is \(0.3\) unit, what is the daily rate of gain in weight? c. A zookeeper wants to bottle-feed an elk calf so as to maintain a daily rate of gain in weight \(G\) of \(0.03\) unit. What must the daily milk-energy intake be? d. The study cited above noted that "the higher levels of milk ingested per unit of body weight are used with reduced efficiency." Explain how the shape of the graph supports this statement.

The population of Mexico: \({ }^{3}\) In 1980 the population of Mexico was about \(67.38\) million. For the years 1980 through 1985 , the population grew at a rate of about \(2.6 \%\) per year. a. Find a formula for an exponential function that gives the population of Mexico. b. Express using functional notation the population of Mexico in 1983 , and then calculate that value. c. Use the function you found in part a to predict when the population of Mexico will reach 90 million.
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