/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Ecologists have studied how a po... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

Ecologists have studied how a population's intrinsic exponential growth rate \(r\) is related to the body weight \(W\) for herbivorous mammals. \({ }^{26}\) In Table \(5.2, W\) is the adult weight measured in pounds, and \(r\) is growth rate per year. $$ \begin{array}{|l|c|c|} \hline \text { Animal } & \text { Weight } W & r \\ \hline \text { Short-tailed vole } & 0.07 & 4.56 \\ \hline \text { Norway rat } & 0.7 & 3.91 \\ \hline \text { Roe deer } & 55 & 0.23 \\ \hline \text { White-tailed deer } & 165 & 0.55 \\ \hline \text { American elk } & 595 & 0.27 \\ \hline \text { African elephant } & 8160 & 0.06 \\ \hline \end{array} $$ a. Make a plot of \(\ln r\) against \(\ln W\). Is it reasonable to model \(r\) as a power function of \(W\) ? b. Find a formula that models \(r\) as a power function of \(W\), and draw a graph of this function.

Short Answer

Expert verified
Yes, it is reasonable to model \(r\) as a power function of \(W\). Use linear regression to find the power function.

Step by step solution

01

Calculate Natural Logarithms

First, calculate the natural logarithm (ln) of the growth rate \(r\) and the body weight \(W\) for each animal in the table. This will help us determine whether \(r\) can be modeled as a power function of \(W\). For each pair, use the formulas \(\ln r\) and \(\ln W\).
02

Create a Data Table

After calculating the natural logarithms, make a new table that includes \(\ln W\) and \(\ln r\) alongside their corresponding animals. This will organize your data for plotting and make the patterns more apparent.
03

Plot \(\ln r\) versus \(\ln W\)

Use the newly created data table to plot \(\ln r\) on the y-axis against \(\ln W\) on the x-axis. Examine the plotted points to determine if they form a straight line, which would suggest that a power function is a reasonable model.
04

Analyze the Plot for Linear Relationship

If the plot of \(\ln r\) versus \(\ln W\) is approximately linear, this suggests that the relationship between \(r\) and \(W\) is a power function. A power function can be expressed as \(r = kW^a\), where \(k\) and \(a\) are constants.
05

Use Linear Regression to Find Constants

Perform a linear regression on the \(\ln r\) versus \(\ln W\) data to find the slope \(a\) and the intercept \(\ln k\) of the line. \(a\) will be the exponent in the power function, while \(k\) is found by exponentiating the intercept: \(k = e^{\text{intercept}}\).
06

Write the Power Function

Using the values obtained from the linear regression, express the power function model for \(r\) in terms of \(W\) as \(r = kW^a\). Insert the specific numerical values you calculated for \(k\) and \(a\).
07

Graph the Power Function

Finally, graph the power function \(r = kW^a\) using a range of \(W\) values that encompass those listed in the original table. Ensure that the graph represents the relationship identified from your regression, showing how \(r\) varies with \(W\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Power Function
A power function is an expression of the form \( r = kW^a \), where \( r \) is the dependent variable, \( W \) is the independent variable, and both \( k \) and \( a \) are constants. In the context of the exercise, ecologists are using a power function to explore the relationship between an animal's body weight and its population growth rate.

This expression suggests that the growth rate depends on weight raised to a certain power \( a \) and multiplied by a constant \( k \). The exponent \( a \) helps determine how significantly the weight affects the growth rate. If the exponent is positive, larger weights result in larger growth rates, provided \( k \) is positive.

By modeling ecological relationships with power functions, scientists can analyze complex biological data and identify patterns. This helps in understanding how different species grow relative to their size, crucial for studying population dynamics and resource allocation in ecosystems. The key to applying power functions successfully is determining the correct values of \( k \) and \( a \) using techniques like linear regression.
Linear Regression
Linear regression is a statistical method used to identify the relationship between variables by fitting a linear equation to the observed data. In this exercise, linear regression helps in finding the parameters \( k \) and \( a \) in the power function \( r = kW^a \).

To use linear regression, the exercise first transforms the data by taking the natural logarithm of both the growth rate \( r \) and the weight \( W \). The goal is to convert the power function into a linear equation: \( \ln r = a \ln W + \ln k \). This means that when we plot \( \ln r \) against \( \ln W \), we should ideally see a straight line if the power function is a good model for the data.

From this plot, linear regression provides the best-fit line. The slope of the line corresponds to the exponent \( a \), and the intercept corresponds to \( \ln k \). By applying linear regression, we determine the most appropriate linear relationship between the log-transformed variables, allowing us to infer the parameters of the original power function.
Natural Logarithm
The natural logarithm, denoted as \( \ln \), is a logarithm to the base of the mathematical constant \( e \), approximately equal to 2.71828. It is widely used in science and engineering due to its mathematical properties, especially when dealing with exponential growth or decay.

In this exercise, the natural logarithm is used to linearize the power function. The relationship between growth rate and weight is not linear, so taking \( \ln r \) and \( \ln W \) allows us to transform the power function \( r = kW^a \) into the linear form of \( \ln r = a \ln W + \ln k \). This transformation is key to employing linear regression to determine \( k \) and \( a \).

Using natural logarithms simplifies the analysis because exponential relationships, which are challenging to work with directly, can often be handled more easily once they are converted into linear relationships. This is why natural logarithms are an essential tool in mathematical modeling of biological processes, economics, and other fields that involve exponential changes.
Population Dynamics
Population dynamics is the study of changes in population sizes and the processes that drive these changes over time. It delves into aspects such as birth rates, death rates, and the growth patterns of populations. Understanding these dynamics is crucial for ecological conservation, resource management, and predicting future trends.

In the exercise, population dynamics are explored by examining the exponential growth rates of different herbivorous mammals relative to their body weights. The idea is that the body weight of an animal can significantly influence its growth rate, potentially affecting how quickly a population can expand under certain environmental conditions.

The power function model discussed is a way to represent this relationship mathematically. Knowing the growth patterns can help ecologists develop more effective conservation strategies. For instance, larger animals with slower growth rates may require different management approaches compared to smaller, faster-growing species.
  • Understanding these patterns allows for predictions about population sustainability.
  • Data can identify critical species traits that influence growth.
  • Mathematical models help in assessing the impact of environmental changes.
Thus, population dynamics, supported by mathematical models like power functions, plays a vital role in managing biodiversity and ecosystem health.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

When a road is being built, it usually has straight sections, all with the same grade, that must be linked to each other by curves. (By this we mean curves up and down rather than side to side, which would be another matter.) It's important that as the road changes from one grade to another, the rate of change of grade between the two be constant. \({ }^{56}\) The curve linking one grade to another grade is called a vertical curve. Surveyors mark distances by means of stations that are 100 feet apart. To link a straight grade of \(g_{1}\) to a straight grade of \(g_{2}\), the elevations of the stations are given by $$ y=\frac{g_{2}-g_{1}}{2 L} x^{2}+g_{1} x+E-\frac{g_{1} L}{2} . $$ Here \(y\) is the elevation of the vertical curve in feet, \(g_{1}\) and \(g_{2}\) are percents, \(L\) is the length of the vertical curve in hundreds of feet, \(x\) is the number of the station, and \(E\) is the elevation in feet of the intersection where the two grades would meet. (See Figure 5.105.) The station \(x=0\) is the very beginning of the vertical curve, so the station \(x=0\) lies where the straight section with grade \(g_{1}\) meets the vertical curve. The last station of the vertical curve is \(x=L\), which lies where the vertical curve meets the straight section with grade \(g_{2}\). Assume that the vertical curve you want to design goes over a slight rise, joining a straight section of grade \(1.35 \%\) to a straight section of grade \(-1.75 \%\). Assume that the length of the curve is to be 500 feet (so \(L=5\) ) and that the elevation of the intersection is \(1040.63\) feet. a. What phrase in the first paragraph of this exercise assures you that a quadratic model is appropriate? b. What is the equation for the vertical curve described above? Don't round the coefficients. c. What are the elevations of the stations for the vertical curve? d. Where is the highest point of the road on the vertical curve? (Give the distance along the vertical curve and the elevation.)

There are 3600 commercial bee hives in a region threatened by African bees. Today African bees have taken over 50 hives. Experience in other areas shows that, in the absence of limiting factors, the African bees will increase the number of hives they take over by \(30 \%\) each year. Make a logistic model that shows the number of hives taken over by African bees after \(t\) years, and determine how long it will be before 1800 hives are affected.

Show that the following data can be modeled by a quadratic function. $$ \begin{array}{|l|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline P(x) & 6 & 5 & 8 & 15 & 26 \\ \hline \end{array} $$

Suppose a certain population is initially absent from a certain area but begins migrating there at a rate of \(v\) individuals per day. Suppose further that this is an animal group that would normally grow at an exponential rate. Then the population after \(t\) days in the new area is given by $$ N=\frac{v}{r}\left(e^{r t}-1\right), $$ where \(r\) is a constant that depends on the species and the environment. If the new location proves unfavorable, then the value of \(r\) may be negative. In such a case, we can rewrite the population function as $$ N=\frac{v}{r}\left(a^{t}-1\right), $$ where \(a\) is less than 1. Under these conditions, what is the limiting value of the population?

Show that the following data cannot be modeled by a quadratic function. $$ \begin{array}{|l|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline P(x) & 5 & 8 & 17 & 38 & 77 \\ \hline \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.