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Chapter 4: Problem 14

Caloric content versus shell length: In 1965 Robert T. Paine \({ }^{19}\) gathered data on the length \(L\), in millimeters, of the shell and the caloric content \(C\), in calories, for a certain mollusk. The table below is adapted from those data. $$ \begin{array}{|c|c|} \hline L \text { =length } & C=\text { calories } \\ \hline 7.5 & 92 \\ \hline 13 & 210 \\ \hline 20 & 625 \\ \hline 24 & 1035 \\ \hline 31 & 1480 \\ \hline \end{array} $$ a. Find an exponential model of calories as a function of length. b. Plot the graph of the data and the exponential model. Which of the data points show a good deal less caloric content than the model would predict for the given length? c. If length is increased by 1 millimeter, how is caloric content affected?

Short Answer

Expert verified
1. An exponential model, \( C = a \cdot b^L \), fits the data. 2. The point at \( L = 24 \) is lower than predicted. 3. Caloric content increases by a factor of \( b \) when \( L \) increases by 1 mm.

Step by step solution

01

Understand the Problem

We need to find an exponential model that expresses calories (C) as a function of shell length (L). This model should fit the given data. We also need to plot the data and the model, and determine the impact on calories if length increases by 1 mm.
02

Fit an Exponential Model

An exponential function can be modeled as \( C = a \cdot b^L \). To find the parameters \(a\) and \(b\), we can use logarithmic transformation, making the equation linear: \( \log C = \log a + L \cdot \log b \).
03

Logarithmic Transformation of Data

Transform the caloric content data using the natural logarithm: \( y = \log C \). Then calculate \( y \) for each given \( C \). For example, for \( C = 92 \), \( y = \log(92) \approx 4.52 \).
04

Perform Linear Regression

Use the transformed data \((L, y)\) to perform linear regression, which will yield the slope \(m\) and intercept \(b\) of the linear line. Here, \(m = \log b\) and \(b = \log a\). Use any statistical software or calculator to perform this regression.
05

Determine Parameters \(a\) and \(b\)

From the linear regression, convert back to the original exponential form using \( a = e^b \) and \( b = e^m \). These will provide you with the specific exponential formula.
06

Plot the Data and Model

Plot the original data points \((L, C)\) and the exponential model on the same graph. Make sure to label axes appropriately. Check for data points that significantly deviate from the model line.
07

Analyze Deviations

Determine which data point(s) show fewer calories than the model predicts by observing which points lie below the curve in the plot.
08

Analyze Impact of Length Increase

To understand how an increase in length affects calories, take the derivative of \( C = a \cdot b^L \), or observe the factor by which calories increase when \( L \) increases by 1: \( C(L+1) = a \cdot b^{L+1} = C(L) \cdot b \). Therefore, caloric content is multiplied by \( b \) when length increases by 1 mm.

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Key Concepts

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

Exponential Functions
Exponential functions are vital in modeling scenarios where growth or decay occurs at a constant rate per interval, and they show a wide range of applications from biology to finance. In our example, the caloric content of a mollusk is modeled as an exponential function of shell length. This means that the amount of calories increases multiplicatively, rather than additive, as the shell length increases. This can be expressed in the formula: \[ C = a \cdot b^L \] where \( C \) stands for the caloric content, \( L \) is the shell length, \( a \) is the initial amount of calories when length is zero, and \( b \) is the growth factor. An important aspect of exponential functions is that they amplify changes in the independent variable quickly. This is evident in our case: a small increase in shell length leads to a relatively large increase in calories.
Data Analysis
Data analysis is the process of inspecting, cleaning, and modeling data to discover useful information and suggest conclusions. Through data analysis, we gain insights into the underlying patterns and relationships within data. In this exercise, analyzing the data on shell length and caloric content helps to fit an accurate model to the data. This includes transforming the original exponential data using logarithms to simplify the relationship into a linear form, which is easier for identifying trends and patterns. Transforming the data means converting the caloric content \( C \) to its logarithmic form \( y = \log C \), allowing us to perform linear regression more effectively. By doing so, we can leverage mathematical techniques to precisely determine the relationship between \( L \) and \( C \), improving the precision of our model.
Linear Regression
Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It helps to identify the linear correlation between the input data once it has been transformed. Through linear regression, we can determine the parameters \( a \) and \( b \) of our exponential model. Once the data is in a logarithmic form, linear regression calculates the best-fit line across the data points, providing a slope \( m \) and intercept \( b \). These values correspond to \( \log b \) and \( \log a \) respectively in the context of our exponential model. By applying exponential functions to these derived parameters, we convert them back to their original non-logarithmic forms: - \( a = e^b \), which is the model's vertical intercept - \( b = e^m \), indicating how calories change with shell length This statistical technique is crucial for predicting outcomes and interpreting how changes in shell length impact caloric content.

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Most popular questions from this chapter

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