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Growth in length of haddock: A study by Riatt showed that the maximum length a haddock could be expected to grow is about 53 centimeters. Let D = D(t) denote the difference between 53 centimeters and the length at age t years. The table below gives experimentally collected values for D. $$ \begin{array}{|c|c|} \hline \text { Age } t & \text { Difference } D \\ \hline 2 & 28.2 \\ \hline 5 & 16.1 \\ \hline 7 & 9.5 \\ \hline 13 & 3.3 \\ \hline 19 & 1.0 \\ \hline \end{array} $$ a. Find an exponential model of \(D\) as a function of \(t\). b. Let \(L=L(t)\) denote the length in centimeters of a haddock at age \(t\) years. Find a model for \(L\) as a function of \(t\). c. Plot the graph of the experimentally gathered data for the length \(L\) at ages \(2,5,7,13\), and 19 years along with the graph of the model you made for \(L\). Does this graph show that the 5 year-old haddock is a bit shorter or a bit longer than would be expected? d. A fisherman has caught a haddock that measures 41 centimeters. What is the approximate age of the haddock?

Step by step solution

01

Identify the Exponential Model for D(t)

Given the data, we'll model the difference, \( D(t) \), as an exponential function of the form \( D(t) = a \, e^{-bt} \). We can use the given data points to determine the values of \( a \) and \( b \). Since \( D(t) \) is the difference between 53 cm and the haddock's length at t, and it gives us the starting point where \( a = D(0) = 53 \). Given that at \( t = 2, D = 28.2 \), we can substitute these into the model and solve for \( b \): \[ 28.2 = 53 \, e^{-2b} \].

02

Solving for the Exponential Constant b

Rearrange the equation \( 28.2 = 53 \, e^{-2b} \) to solve for \( b \). Compute \( e^{-2b} = \frac{28.2}{53} \), then take the natural logarithm of both sides to find \( b \): \[ -2b = \ln\left(\frac{28.2}{53}\right) \] leading to \[ b = -\frac{\ln(\frac{28.2}{53})}{2} \]. Compute the value of \( b \).

03

Finding the Length Function L(t)

The length \( L(t) \) can be found from the relationship \( L(t) = 53 - D(t) \). Substitute the exponential model for \( D(t) \) into this equation to find \( L(t) \): \[ L(t) = 53 - 53 \, e^{-bt} \]. This gives \( L(t) = 53(1 - e^{-bt}) \).

04

Plotting the Data and Model

Plot the experimentally gathered data points \((t, L(t))\) for ages 2, 5, 7, 13, and 19. Also plot the model \( L(t) = 53(1 - e^{-bt}) \) using the calculated \( b \). By comparing the 5-year-old haddock data point with the model, determine if it is longer or shorter than expected.

05

Estimating the Age of a 41 cm Haddock

Using the model \( L(t) = 53(1 - e^{-bt}) \), set \( L(t) = 41 \) to find the haddock's age: \[ 41 = 53(1 - e^{-bt}) \ e^{-bt} = 1 - \frac{41}{53} \ -bt = \ln\left(1 - \frac{41}{53}\right) \]. Solve for \( t \) using the value of \( b \) found earlier.

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Key Concepts

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

Haddock Growth

Understanding how haddock grow is crucial for predicting their size at different ages. Haddock, like many fish, have a maximum growth capacity, reaching a certain length by the adult stage. In this scenario, research has shown that haddock can grow up to 53 centimeters. However, they do not reach this length immediately; it gradually increases over time. By studying the growth rates, we can determine the difference in length at various ages, which provides insight into their overall growth pattern. Understanding this growth is vital for fisheries management to ensure sustainable fishing practices.

Exponential Function

The exponential function is a mathematical expression used to model various natural phenomena, including growth processes. In our study of haddock growth, this function helps model how the difference in length changes with age. The general form of an exponential function is \( D(t) = a \, e^{-bt} \), where:

• \( D(t) \) represents the difference in length at age \( t \).
• \( a \) is the initial difference in length when age \( t = 0 \).
• \( e \) is the base of the natural logarithm.
• \( b \) is a constant that represents the growth rate.

Such functions characterize situations where growth rate diminishes over time, explaining why younger haddock grow faster than older ones. The exponential decrease in difference illustrates how they approach the maximum length of 53 centimeters.

Mathematical Modeling

Mathematical modeling is a powerful tool that converts complex real-world situations into understandable mathematical expressions. For haddock growth, we use an exponential model to predict changes in their length over time. This approach allows us to create a model \( L(t) = 53(1 - e^{-bt}) \), where \( L(t) \) represents the haddock's length at any given age \( t \). Mathematical modeling bridges the gap between theoretical predictions and practical understanding, providing a reliable way to forecast future outcomes based on past and present data. By refining these models, we improve our ability to manage marine resources effectively.

Length Prediction

Predicting the length of haddock at different ages helps in understanding both natural growth and managing fish stocks. Length prediction models use collected data to identify growth trends, informing decisions on catch limits and sizes.The formula \( L(t) = 53(1 - e^{-bt}) \) allows us to estimate a haddock's length at any given age. By analyzing past length data and calculating the growth rate constant, \( b \), we offer insights into future growth. For example, if you catch a haddock measuring 41 centimeters, you can reverse-engineer the equation to estimate its age. This predictive power not only informs science but supports sustainable practices, ensuring future generations of haddock are abundant.