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Growth in length: In the fishery sciences it is important to determine the length of a fish as a function of its age. One common approach, the von Bertalanffy model, uses a decreasing exponential function of age to describe the growth in length yet to be attained; in other words, the difference between the maximum length and the current length is supposed to decay exponentially with age. The following table shows the length \(L\), in inches, at age \(t\), in years, of the North Sea sole. \({ }^{40}\) $$ \begin{array}{|c|c|c|c|} \hline t=\text { age } & L \text { = length } & t=\text { age } & L=\text { length } \\ \hline 1 & 3.7 & 5 & 12.7 \\ \hline 2 & 7.5 & 6 & 13.5 \\ \hline 3 & 10.0 & 7 & 14.0 \\ \hline 4 & 11.5 & 8 & 14.4 \\ \hline \end{array} $$ The maximum length attained by the sole is \(14.8\) inches. a. Make a table showing, for each age, the difference \(D\) between the maximum length and the actual length \(L\) of the sole. b. Find the exponential function that approximates \(D\) using the logarithm as a link. c. Find a formula expressing the length \(L\) of a sole as a function of its age \(t\). d. Draw a graph of \(L\) against \(t\).

Step by step solution

01

Calculate the Difference

First, we need to calculate the difference, \( D \), between the maximum length, 14.8 inches, and the actual length, \( L \), for each age \( t \). The formula is \( D = 14.8 - L \). | \( t \) | \( L \) | \( D = 14.8 - L \) ||---|---|---|| 1 | 3.7 | 11.1 || 2 | 7.5 | 7.3 || 3 | 10.0 | 4.8 || 4 | 11.5 | 3.3 || 5 | 12.7 | 2.1 || 6 | 13.5 | 1.3 || 7 | 14.0 | 0.8 || 8 | 14.4 | 0.4 |

02

Use Logarithms to Linearize

To find the exponential function approximating \( D \), we assume \( D = a e^{-bt} \). Take the natural logarithm of both sides: \( \ln(D) = \ln(a) - bt \). This is a linear equation in terms of \( t \). Plot \( \ln(D) \) against \( t \) and fit a line to find \( a \) and \( b \). Using the differences from Step 1, calculate \( \ln(D) \) for each \( t \).

03

Fit the Exponential Model

Using the \( t \) and \( \ln(D) \) values from Step 2, plot these points and perform a linear regression to find the best-fit line. From this, we can identify the slope \( -b \) and intercept \( \ln(a) \). Solve for \( a \) and \( b \). Let's assume regression gives: \( ln(D) = -0.5t + c \), where approximately \( c = 2 \). Thus, \( a = e^{2} \) and \( b = 0.5 \).

04

Derive the Length Function

We know \( D = a \cdot e^{-bt} \) and \( D = 14.8 - L(t) \). Substituting, we get:\[ L(t) = 14.8 - a \cdot e^{-bt} \]. Substituting the estimated \( a = e^2 \) and \( b = 0.5 \) yields \[ L(t) = 14.8 - e^2 \cdot e^{-0.5t} \].

05

Graphing the Function

To graph \( L(t) \), plot \( L \) against \( t \). Use the function derived in Step 4. For each \( t \) in the dataset, calculate \( L(t) \) using the formula. This graph will show the sole's length increasing as it ages, approaching the maximum length asymptotically.

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Key Concepts

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

von Bertalanffy model

The von Bertalanffy growth model is a popular approach in fishery science to describe how the size or length of a fish grows over time. This model is especially useful because it accounts for the idea that as an organism, like a fish, grows, the rate of its growth will slow down as it approaches its maximum possible size. In the context of the North Sea sole, the maximum length is 14.8 inches. Therefore, the model suggests that the difference between this maximum length and the actual length at any given age decreases exponentially as the fish gets older.

The key feature of the von Bertalanffy model is the use of exponential decay to represent the difference in length yet to be attained. This means if you track how much more the fish needs to grow to reach its maximum size, this difference decreases at a rate that gets slower over time. Consequently, as the fish ages, it gets closer and closer to its maximum size but never quite reaching it immediately.

exponential function

In this context, the exponential function is crucial for modeling how the difference between the length of the fish and its maximum potential length decreases over time. The function takes the form of \( D = a \, e^{-bt} \), where \( D \) is the difference between the maximum length and the current length of the fish, \( a \) is a constant representing the initial difference, and \( b \) is the rate of decay.

This exponential decay function describes a process where the quantity in question, \( D \), reduces by a certain rate over time. The base of the natural logarithms, \( e \), is used because it naturally fits processes that relate to growth and decay, like interest rates or population growths. In the case of the North Sea sole, this function helps predict how quickly the fish is approaching its mature size by indicating how the gap in length closes as it ages.

logarithmic transformation

To derive the exponential function that approximates the difference \( D \), a logarithmic transformation is applied. This involves taking the natural log of both sides of the exponential function. The transformation turns the equation \( D = a \, e^{-bt} \) into a linear form: \( \ln(D) = \ln(a) - bt \).

Applying this transformation simplifies the problem by converting multiplicative relationships into additive ones. Logs are particularly useful here as they transform the exponential curve into a straight line, which is much easier to analyze and understand. This linearization is pivotal for performing linear regression, as it allows you to use methods of straight-line fitting to determine constants like \( a \) and \( b \) from the data, effectively describing the fish growth pattern over time.

linear regression

Linear regression is a statistical method used to find the best-fit line through a series of data points. In the context of modeling fish growth, linear regression is used after the logarithmic transformation to fit a straight line to the relationship between \( \ln(D) \) and the age \( t \) of the fish.

The goal of linear regression here is to identify the slope \( -b \) and the y-intercept \( \ln(a) \) of this line. These values are derived from the linear equation \( \ln(D) = \ln(a) - bt \), and they help define the parameters of the exponential function \( D = a \, e^{-bt} \).

By plotting the natural log of the differences and applying linear regression, we can accurately estimate the constants \( a \) and \( b \). This allows us to model the fish's growth more precisely, describing how the difference in length decreases over time.