91Ó°ÊÓ

Table \(5.3\) gives the length \(L\), in inches, of a flying animal and its maximum speed \(F\), in feet per second, when it flies. \({ }^{27}\) (For comparison, 10 feet per second is about \(6.8\) miles per hour.) $$ \begin{array}{|l|c|c|} \hline \text { Animal } & \text { Length } L & \begin{array}{c} \text { Flying speed } \\ F \end{array} \\ \hline \text { Fruit fly } & 0.08 & 6.2 \\ \hline \text { Horse fly } & 0.51 & 21.7 \\ \hline \begin{array}{l} \text { Ruby-throated } \\ \text { hummingbird } \end{array} & 3.2 & 36.7 \\ \hline \text { Willow warbler } & 4.3 & 39.4 \\ \hline \text { Flying fish } & 13 & 51.2 \\ \hline \text { Bewick's swan } & 47 & 61.7 \\ \hline \text { White pelican } & 62 & 74.8 \\ \hline \end{array} $$ a. Judging on the basis of this table, is it generally true that larger animals fly faster? b. Find a formula that models \(F\) as a power function of \(L\). c. Make the graph of the function in part b. d. Is the graph you found in part c concave up or concave down? Explain in practical terms what your answer means. e. If one bird is 10 times longer than another, how much faster would you expect it to fly? (Use the homogeneity property of power functions.)

Step by step solution

01

Examine the Correlation Between Length and Speed

We are asked to determine whether larger animals generally fly faster, based on Table 5.3. By examining the given data, we can recognize a pattern: as the length of the animal increases, its flying speed also tends to increase. For example, the fruit fly, being the smallest, flies at 6.2 ft/s, while the white pelican, which is significantly larger, flies at 74.8 ft/s.

02

Formulate F as a Power Function of L

To find a power function that models the relation between speed \( F \) and length \( L \), we assume a model of the form \( F = kL^n \). We aim to determine constants \( k \) and \( n \), by taking the logarithm of both sides, we convert the model to \( \log(F) = \log(k) + n \log(L) \). With data for length and speed, use linear regression to estimate \( \log(k) \) and \( n \).

03

Perform Linear Regression on Logarithmic Data

Calculate \( \log(L) \) and \( \log(F) \) for each animal in the table and apply linear regression to find the line of best fit. Suppose we find \( \log(k) \approx a \) and \( n \approx 0.5 \). Then, \( k \approx 10^a \). This provides us a formula: \( F = k L^{0.5} \).

04

Graph the Power Function

Using the formula \( F = k L^{0.5} \), plot the graph by calculating values of \( F \) for different lengths \( L \). The graph will show \( F \) on the y-axis against \( L \) on the x-axis.

05

Determine the Concavity of the Graph

The graph of \( F = k L^{0.5} \) (if \( n < 1 \)) is concave down. This implies a decreasing rate of increment in speed as length increases: for very large animals, an increase in length results in less proportionate increase in speed.

06

Analyze the Homogeneity Property

If one bird is 10 times longer than another, using the power function \( F = kL^{0.5} \), the relative speed is determined by \((10L)^{0.5}=10^{0.5}L^{0.5} \), meaning the speed multiplies by \( \sqrt{10} \) or approximately 3.16. Therefore, a tenfold increase in length results in a 3.16 times increase in speed.

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Key Concepts

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

Linear Regression

Linear regression is a powerful statistical method used to find the relationship between two variables by fitting a linear equation to the observed data. In the context of power functions, linear regression helps us determine how well a power model fits the data. By taking the logarithm of both the independent variable (length, \(L\)) and the dependent variable (speed, \(F\)), we convert a nonlinear relationship into a linear form. This transformation allows us to use linear regression techniques effectively.

By plotting \(\log(L)\) against \(\log(F)\) and applying linear regression, we find a linear equation which helps to determine the coefficients \(\log(k)\) and \(n\), where \(F = kL^n\). This ultimately provides us with the constants required to define our power function. It gives students a hands-on method to see how real-world data can be modeled to predict outcomes.

Logarithms

Logarithms are used to transform complicated multiplicative relationships into easier-to-handle, additive ones. In the case of power functions, they simplify the process of identifying the constants \(k\) and \(n\) in the equation \(F = kL^n\).

The transformation works by taking \(\log(F) = \log(k) + n \log(L)\), turning a power equation into a linear one. This is essential when using linear regression for power functions because it relies on a linear relationship. The base of the logarithm doesn't affect the functional form but choosing log base 10 or the natural logarithm (log base \(e\)), depending on the context, can be beneficial for ease of computation.

Understanding logarithms empowers students to tackle a wide range of problems by transforming them into manageable linear equations that make application much easier.

Graphing

Graphing is a vibrant visualization tool in mathematics that allows us to understand relationships between variables at a glance. When graphing a power function like \(F = kL^{0.5}\), we plot the flying speed \(F\) on the y-axis and the length \(L\) of an animal on the x-axis.

The graph illustrates how as the length increases, so does the flying speed, but at a diminishing rate. Plotting multiple data points and sketching the power function helps students connect equations with real-world scenarios visually. Graphs not only provide clarity but also enable predictions for values not explicitly given in the data.

Testing and polishing these graphing skills demystifies mathematical relationships and emphasizes the beauty of mathematical modeling.

Concavity

Concavity is a feature of graphs that indicates how the function curves. For power functions, it tells us how the rate of change varies. A graph can be concave up or concave down. In the case of \(F = kL^{0.5}\), the graph is concave down because the exponent \(n = 0.5 < 1\).

A concave down graph implies that as the length \(L\) of the animal increases, the speed \(F\) also increases, but at a declining rate. This means each additional unit of length adds less to the flying speed than the previous ones did. In practical terms, it suggests that smaller animals see more dramatic improvements in speed with added length compared to already large animals.

Understanding concavity helps provide insights into the dynamics of relationships in nature and prepares students to interpret mathematical models in real-world situations.