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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.16 $$ \begin{array}{|c|c|c|c|} \hline t=\text { age } & L=\text { length } & t=\text { age } & L=\text { length } \\ \hline 1 & 3.7 & 5 & 12.7 \\ \hline 2 & 7.5 & 6 & 13.5 \\ \hline 3 & 10.0 & 7 & 14.0 \\ \hline 4 & 11.5 & 8 & 14.4 \\ \hline \end{array} $$ 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. 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. e. If a sole is 11 inches long, how old is it? f. Calculate L(9) and explain in practical terms what your answer means.

Step by step solution

01

Calculate Length Difference

Create a table listing the difference \( D = 14.8 - L \) for each age. For example, for age \( t = 1 \), \( D = 14.8 - 3.7 = 11.1 \). Follow this process for ages 2 through 8 to complete the table.

02

Determine the Exponential Function for D

Assume the form \( D = D_0 e^{-kt} \). Using the differences from the table, solve for \( D_0 \) and \( k \) by fitting an exponential curve to the data points \((t, D)\). This can be done using regression analysis tools or a graphing calculator.

03

Derive the Length Function in terms of Age

From the relation \( L = 14.8 - D \), substitute the expression for \( D \): \( L = 14.8 - D_0 e^{-kt} \), giving \( L(t) \) as a function of age \( t \).

04

Graph Length vs. Age

Plot \( L(t) \) against \( t \) using the derived exponential function. This can be done manually or with graphing software by plotting age on the x-axis and length on the y-axis.

05

Determine Age for 11-Inch Length

Set \( L = 11 \) in the equation \( 14.8 - D_0 e^{-kt} = 11 \). Solve for \( t \) to find the age at which a sole is 11 inches long using the derived parameters \( D_0 \) and \( k \).

06

Calculate L(9) and Interpret

Substitute \( t = 9 \) into the length function to find \( L(9) \). For example, if the calculation results in \( L(9) = 14.6 \), this means that at age 9, a North Sea sole is approximately 14.6 inches long.

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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 model is a growth model often used in biology and fishery sciences to describe how organisms grow over time. It provides a mathematical representation of growth that can be generalized to many species. The model uses an exponential function to describe the decrease in growth as an organism approaches its maximum size. In simpler terms, this model assumes that the growth rate of an organism slows down as it gets closer to its mature size.

The formula commonly used in the von Bertalanffy model is expressed as a decreasing exponential function:

• \( D = D_0 e^{-kt} \)

Here, \( D \) is the difference between maximum possible length and observed length, \( D_0 \) is the initial difference when the organism is young, and \( k \) is a rate constant that specifies how quickly growth slows down. This model is particularly useful because it gives scientists a predictable way to assess an organism’s growth over its lifespan.

fishery sciences

Fishery sciences focus on managing and understanding fish stocks, ensuring sustainable practices, and maintaining aquatic ecosystems. An integral part of fishery science is studying the growth patterns of fish, as understanding these patterns helps in making informed decisions about managing fish populations.

One key aspect in fisheries is determining the length of fish as they age, which is critical for:

• Establishing growth rates across different environments.
• Predicting future stock sizes based on growth trends.
• Setting catch limits to avoid overfishing.

Models like the von Bertalanffy are used extensively in fisheries to predict how fish populations grow over time. By knowing how length relates to age, fisheries can estimate biomass and evaluate the health of the fish stocks. This information is essential for crafting policies that aim to preserve fish populations for future generations.

exponential function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. This type of function often describes processes where growth rate is proportional to the current quantity. In biology and fishery sciences, exponential functions are crucial because many natural phenomena, like growth and decay processes, can be described using these functions.

In the context of the von Bertalanffy model, the exponential function is used to show how the difference between current fish length and its maximum possible length decreases over time:

• \( D = D_0 e^{-kt} \)

Here, \( e \) is the base of the natural logarithm, which is approximately equal to 2.718. \( k \) determines the rate of growth decay, meaning how fast the fish stops growing. Using exponential functions allows for a more accurate representation of growth as it naturally slows down with time.

age-length relationship

The age-length relationship in fish describes how the size of a fish relates to its age. This relationship is essential for understanding fish growth patterns, making it one of the focal points in fishery research. By analyzing age-length data, scientists can assess:

• How fast a fish species grows during different life stages.
• The overall health and condition of fish populations.
• Factors affecting fish growth, such as environment and genetics.

The von Bertalanffy growth model illustrates this relationship by providing a mathematical framework that predicts how fish length increases with age. For instance, in the given exercise, fish length data across different ages helps form a function, \( L(t) = 14.8 - D_0 e^{-kt} \), which precisely describes the growth pattern of the North Sea sole. Such models assist in determining the age of a fish that has reached a particular length, informing studies on maturation, spawning, and population dynamics.