91Ó°ÊÓ

This is a continuation of Exercise 5. Table \(5.4\) gives the length \(L\), in inches, of a swimming animal and its maximum speed \(S\), in feet per second, when it swims. a. Find a formula for the regression line of \(\ln S\) against \(\ln L\). (Round the slope to one decimal place.) b. Find a formula that models \(S\) as a power function of \(L\). On the basis of the power you found, what special type of power function is this? $$ \begin{array}{|l|c|c|} \hline \text { Animal } & \text { Length } L & \begin{array}{c} \text { Swimming } \\ \text { speed } S \end{array} \\ \hline \text { Bacillus } & 9.8 \times 10^{-5} & 4.9 \times 10^{-5} \\ \hline \text { Paramecium } & 0.0087 & 0.0033 \\ \hline \text { Water mite } & 0.051 & 0.013 \\ \hline \text { Flatfish larva } & 0.37 & 0.38 \\ \hline \text { Goldfish } & 2.8 & 2.5 \\ \hline \text { Dace } & 5.9 & 5.7 \\ \hline \text { Adélie penguin } & 30 & 12.5 \\ \hline \text { Dolphin } & 87 & 33.8 \\ \hline \end{array} $$ c. Add the graph of \(S\) against \(L\) to the graph of \(F\) that you drew in Exercise \(5 .\) d. Is flying a significant improvement over swimming if an animal is 1 foot long (the approximate length of a flying fish)? e. Would flying be a significant improvement over swimming for an animal 20 feet long? f. A blue whale is about 85 feet long, and its maximum speed swimming is about 34 feet per second. Judging on the basis of these facts, do you think the trend you found in part b continues indefinitely as the length increases?

Step by step solution

01

Transform the Data Using Natural Logarithms

For each pair of length \( L \) and speed \( S \), calculate \( \ln(L) \) and \( \ln(S) \). This will transform the data for a linear regression of \( \ln(S) \) against \( \ln(L) \).

02

Calculate the Linear Regression Coefficients

Using the transformed data from Step 1, compute the slope and intercept of the regression line that models \( \ln(S) = a + b \ln(L) \). Use a statistical calculator or software to perform the regression. Round the slope \( b \) to one decimal place.

03

Derive the Power Function Model

Convert the regression model \( \ln(S) = a + b\ln(L) \) back into a power function form \( S = kL^b \) by using exponentiation. Here, \( k = e^a \), where \( e \) is the base of natural logarithms.

04

Identify the Type of Power Function

Examine the value of \( b \) obtained in Step 2. If \( b \) is close to 1, it becomes a linear function. If \( b > 1 \), it indicates an accelerating function, and if \( 0 < b < 1 \), it suggests a decelerating function.

05

Graph the Relation and Compare

Plot the derived power function \( S = kL^b \) against actual points for confirming the fit visually. Add this graph to the existing graph of function \( F \) from Exercise 5.

06

Evaluate Impact of Flying at 1 Foot Length

Analyze whether a flying animal at 1-foot length shows significant speed over a swimming counterpart by comparing the derived swimming speed from the function with known speeds of flying animals.

07

Assess Improvement at 20 Feet Length

Check if flying significantly improves speed over swimming by doing a similar comparison for a 20-foot-long animal.

08

Analyze Speed for 85-foot Length

Compare the function's prediction for \( S \) with the actual speed of a blue whale. Consider whether the trend of the power function holds as \( L \) increases further.

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Key Concepts

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

Power Function

In mathematics, a power function is simply an algebraic expression of the form \( S = kL^b \). This function describes a relationship where the dependent variable (in this case, the swimming speed \( S \)) is proportional to a fixed power of the independent variable (the length \( L \)). This means that instead of changing linearly, \( S \) accelerates, decelerates, or remains linear depending on the value of the exponent \( b \).

• If \( b = 1 \), the function is linear, meaning \( S \) increases proportionally with \( L \).
• If \( b > 1 \), the function is accelerating, implying that \( S \) increases more rapidly as \( L \) grows.
• If \( 0 < b < 1 \), the function is decelerating, so \( S \) increases but at a diminishing rate as \( L \) increases.

To derive a power function from a set of data, one often fits the data using a transformed linear regression model in the logarithmic form \( \ln(S) = a + b\ln(L) \), before converting back to the power function form.

Natural Logarithm Transformation

Natural logarithm transformation is a technique used to linearize data that exhibit a power relationship. When you have a non-linear relationship between variables, the relationship can often be expressed in a simpler linear fashion by taking the natural logarithm (often denoted as \( \ln \)) of the data.

The transformation involves changing the original data \((L, S)\) to \((\ln(L), \ln(S))\). By applying this transformation, the equation \( S = kL^b \) becomes \( \ln(S) = \ln(k) + b\ln(L) \). The logarithm acts as a mathematical flattener, allowing non-linear data to be treated with linear methods like regression.

This makes it easier to perform statistical analyses, such as determining the slope \( b \) and intercept \( a \) of the regression, which are key to understanding the relationship between \( S \) and \( L \). Once this linear regression analysis is complete, the results enable us to revert to the original form of the equation, thus offering an intuitive understanding of the model.

Linear Regression

Linear regression is a statistical method for modeling the relationship between two variables by fitting a linear equation to observed data. When working with transformed logarithmic data, it helps in estimating the parameters of the model; these are typically the slope \( b \) and the intercept \( a \).

• Slope Calculation: The slope \( b \) determines how much \( \ln(S) \) changes with a unit change in \( \ln(L) \). In our power function model, it governs whether the relationship is accelerating, linear, or decelerating.
• Intercept Calculation: The intercept \( a \) represents where the line crosses the y-axis when all other variables are equal to zero. In terms of the power function model, this corresponds to \( \ln(k) \), enabling us to calculate \( k \) using \( e^a \).

The purpose of linear regression in this context is to ensure that the power relationship between swimming speed and length is accurately captured and can be easily interpreted and predicted through the inference drawn from transformed data.

Graphing Functions

Graphing functions is an essential aspect of understanding relationships between variables visually. In the context of this exercise, plotting functions involves graphing the original data points \((L, S)\) and the derived power function \( S = kL^b \) to confirm the fit and accuracy visually.

By plotting, you can see how well the power function line matches the actual data points. A good visual alignment suggests that the function accurately captures the trend in the data. Consider these steps:

• Plot the original data points to understand the distribution and range of \( L \) and \( S \).
• Add the power function graph to this plot to visually verify how well it fits the points.
• Analyze the graph for any significant deviations to understand potential limitations or errors.

Additionally, graphing can be crucial for comparing relationships across different variables, such as comparing the swimming speed with the flying speed of animals, which helps in greater comparative analysis and deduction.