Problem 22 - College Algebra with Modeling and Visualization

Question

Heavier birds tend to have a longer wing span than smaller birds. For one species of bird, the table lists the length \(L\) of the bird's wing span in feet if the bird weighs w pounds. $$\begin{array}{rrrrr}w(\mathrm{b}) & 0.22 & 0.88 & 1.76 & 2.42 \\\L(w)(\mathrm{ft}) & 1.38 & 2.19 & 2.76 & 3.07\end{array}$$ (a) Find a function that models the data. (b) Graph \(L\) and the data. (c) What weight corresponds to a wing span of 2 feet?

Step-by-Step Solution

Answer

The approximate function is \( L(w) = 0.768w + 1.22 \). A weight of about 1.02 pounds corresponds to a wing span of 2 feet.

Step 1: Analyze the Data

We have two sets of corresponding values, weight \( w \) in pounds and wing span \( L \) in feet for particular birds. We need to identify whether there is a linear or non-linear relationship between these variables.

Step 2: Test for Linearity

Using the given data, check if the change in wing span \( L \) is proportional to the change in weight \( w \). Calculate differences between consecutive \( w \) and \( L \) values to see if the ratio \( \frac{\Delta L}{\Delta w} \) is constant.

Step 3: Calculate Slopes

Find the slopes between each pair of points:- From (0.22, 1.38) to (0.88, 2.19): \( \frac{2.19 - 1.38}{0.88 - 0.22} = \frac{0.81}{0.66} = 1.227 \)- From (0.88, 2.19) to (1.76, 2.76): \( \frac{2.76 - 2.19}{1.76 - 0.88} = \frac{0.57}{0.88} = 0.648 \)- From (1.76, 2.76) to (2.42, 3.07): \( \frac{3.07 - 2.76}{2.42 - 1.76} = \frac{0.31}{0.66} = 0.47 \)\.

Step 4: Determine the Type of Function

Since the slope \( \frac{\Delta L}{\Delta w} \) is not consistent, the data does not perfectly fit a linear model. A non-linear regression or piecewise function might be necessary, but a rough linear approximation will be used for simplicity.

Step 5: Approximate Linear Function

We choose a rough linear function for simplicity. Using the first and last points to find a linear approximation: \( L = mw + b \). Calculate \( m = \frac{3.07 - 1.38}{2.42 - 0.22} = \frac{1.69}{2.2} \approx 0.768 \) and using point (0.22, 1.38) for \( b \), \( 1.38 = 0.768 \times 0.22 + b \Rightarrow b \approx 1.22 \). Model is \( L(w) = 0.768w + 1.22 \).

Step 6: Graph the Function and Data

Plot the points (0.22, 1.38), (0.88, 2.19), (1.76, 2.76), and (2.42, 3.07) on a graph. Then draw the line \( L = 0.768w + 1.22 \) to see how well it fits.

Step 7: Find Weight for Wing Span of 2 feet

Substitute \( L = 2 \) into the linear function \( L = 0.768w + 1.22 \) to solve for \( w \):\[ 2 = 0.768w + 1.22 \]\[ 0.78 = 0.768w \]\[ w \approx \frac{0.78}{0.768} \approx 1.02 \text{ pounds}\].

Key Concepts

Linear Regression

In the world of data analysis, linear regression is a popular technique used to model the relationship between two sets of variables. It helps us to understand how one variable, known as the independent variable, influences another, the dependent variable. In this exercise, we are trying to determine if there is a linear correlation between the weight of the birds and their wing span.

The term 'linear' implies that the relationship can be represented as a straight line, summarized by the equation \( L = mw + b \), where \( L \) is the wing span, \( w \) is the weight, \( m \) represents the slope, and \( b \) is the y-intercept. The slope \( m \) tells us how much \( L \) changes for each unit change in \( w \). If the change is more or less consistent, then linear regression might be suitable for this data.

In this exercise, we observed that the slope \( \frac{\Delta L}{\Delta w} \) is not consistent across different weight intervals. Therefore, although the relationship appears to be somewhat linear, it does not exactly fit a straight line model. Nonetheless, using a linear approximation, we can still model the underlying pattern to a simple linear equation, simplifying predictions and interpretations.

Data Analysis

When we analyze data, like the wing span and weight of birds, we look at the data for specific patterns or relationships. This process involves organizing, interpreting, and presenting the data. In the original exercise, we collect the birds' wing spans and weights in a tabular form to identify any discernible pattern or trend between these two variables.

The goal is to find significant trends, such as how the wing span changes as the weight increases. The identification of these patterns is crucial for making predictions and informed decisions, like determining potential wing span given a specific weight. In this context, it's essential to consider the possible presence of non-linearity within the data, which can reveal complex relationships that a simple linear model might miss.

During data analysis, it's important to calculate the differences and slopes between the consecutive pairs of data points. These calculate changes may indicate whether data trends are consistent and can simplify predictions. It gives us better insight into whether a linear model can be effective or if a more complex approach is needed.

Function Approximation

Function approximation helps to create simpler functions that are close representations of more complex real-world data. It enables us to make estimations or predictions when dealing with practical scenarios. In this exercise, we approximated a linear function to model bird weights and wing spans.

Although the computed slopes varied, and a perfect linear relationship doesn't exist, we still use a function approximation to represent the data comprehensively with a simplified model. We chose the largest data points, from smallest to largest, to construct a linear equation as an approximation. This allows us to derive a general notion of the underlying trend.

Such approximations are particularly useful in making quick predictions without needing complex calculations. For instance, finding a corresponding weight for a given wing span using the formula \( L(w) = 0.768w + 1.22 \). By substituting the wing span value of 2 feet, we accurately obtain an approximate bird weight of 1.02 pounds, showcasing the power of a well-approximated function to inform and simplify solutions in applied contexts.