91Ó°ÊÓ

In the study by Beverton and Holt \({ }^{42}\) described in Example 5.13, information is also provided on haddock. We have adapted it for this exercise. We measure length in inches, weight in pounds, and age \(t\) in years. a. It was found that haddock grow to a maximum length of about 21 inches. Furthermore, the difference \(D=D(t)\) between maximum length and length \(L=L(t)\) at age \(t\) is an exponential function. If initially a haddock is 4 inches long, find the initial value of \(D\). b. It was found that a 6-year-old haddock is about \(15.8\) inches long. Find a formula that gives \(L\) in terms of \(t\). c. The weight \(W\) in pounds of a haddock can be calculated from its length \(L\) using $$ W=0.000293 L^{3} \text {. } $$ Use function composition to find a formula for weight \(W\) as a function of age \(t\). d. After \(t\) years, the total number of fish in the cohort is given by the exponential function $$ N=1000 e^{-0.2 t} \text {. } $$ The total biomass of the cohort is the number of fish times the weight per fish. Find a formula for the total biomass \(B\) of the cohort as a function of the age of the cohort. e. How old is the cohort when biomass is at a maximum? (Consider ages up to 10 years.)

Step by step solution

01

Define the Difference Function

Given the maximum length of 21 inches, the difference function is defined as \( D(t) = 21 - L(t) \), where \( L(t) \) represents the length at age \( t \). Initially, the length is 4 inches, so \( D(0) = 21 - 4 = 17 \).

02

Determine Length Formula

To find the formula for length \( L(t) \), we know that \( D(t) = 21 - L(t) \) is exponential. Let's express \( D(t) = D_0 e^{-kt} \), where \( D_0 = 17 \) represents the initial difference and \( k \) is a constant. We solve for \( L(t) = 21 - 17e^{-kt} \). At \( t = 6 \), \( L(6) = 15.8 \). Substituting this into the formula gives \( 15.8 = 21 - 17e^{-6k} \) for \( k \).

03

Calculate the Constant k

Rearranging the equation \( 15.8 = 21 - 17e^{-6k} \) gives \( 17e^{-6k} = 5.2 \). Dividing both sides by 17 gives \( e^{-6k} = \frac{5.2}{17} \). Taking the natural logarithm of both sides gives \( -6k = \ln\left(\frac{5.2}{17}\right) \). Solving for \( k \) gives \( k = -\frac{1}{6}\ln\left(\frac{5.2}{17}\right) \).

04

Complete the Length Formula

With \( k \) calculated, the length formula becomes \( L(t) = 21 - 17e^{-kt} \) where \( k = -\frac{1}{6}\ln\left(\frac{5.2}{17}\right) \).

05

Establish the Weight Function in Terms of Age

Using the function composition with the weight formula \( W = 0.000293 L^3 \), substitute \( L(t) = 21 - 17e^{-kt} \) to get \( W(t) = 0.000293 (21 - 17e^{-kt})^3 \).

06

Compute Biomass Function in Terms of Age

The biomass \( B \) is the product of the number of fish \( N = 1000e^{-0.2t} \) and weight \( W(t) \). Therefore, \( B(t) = 1000e^{-0.2t} \cdot 0.000293 (21 - 17e^{-kt})^3 \).

07

Determine When Biomass is Maximum

To find when biomass is maximum, take the derivative of \( B(t) \) with respect to \( t \), equate to zero and solve for \( t \). This requires differentiating \( B(t) \) and finding critical points, then evaluating them in terms of the given constraint (up to 10 years).

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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 situations where growth or decay happens continuously and at a rate proportional to the current amount. One common form is \( f(t) = a e^{kt} \), landing it directly applicable for many natural processes. In our exercise, we explore how the difference in length of a haddock over time forms an exponential function. The difference in length \( D(t) \) decays as the fish grows, modeled by\[ D(t) = D_0 e^{-kt}, \]where \( D_0 \) is the initial length difference, and \( k \) is a constant reflecting the rate of closing the gap between maximum and present lengths.

Key aspects of working with exponential functions:

• Initial Value: The initial value \( D_0 \) is where the function begins. It's particularly useful for setting up the model.
• Rate of Change: The exponent \( -kt \) expresses how rapidly the quantity is changing.

Exponential decay models are apparent in natural processes because they continuously adjust the rate of change based on the current state of the quantity, producing smooth and realistic growth curves.

Biomass Calculation

Biomass calculation important for understanding the mass of all individuals in a population or group. In this context, it combines the weight of each fish in the cohort with the number of individuals.
The formula for biomass expresses it as the product of two functions:

• The number of fish, an exponential function: \( N(t) = 1000 e^{-0.2t} \)
• The weight per fish: \( W(t) = 0.000293 L(t)^3 \)

Together, these give the total biomass \( B(t) \), ensuring a dynamic measure that accounts for both individual growth and population decline\[ B(t) = N(t) \, W(t) = 1000 e^{-0.2t} \times 0.000293 (L(t))^3. \]Considerations for calculating biomass include:

• Population Change: As \( t \) increases, \( N(t) \) decreases, reflecting natural mortality or migration.
• Individual Growth: \( L(t) \), tied to age, affects the calculation as fish grow larger and heavier over time.

Calculating biomass like this is common in ecological studies and fishery management, providing insight into various ecological dynamics and resource estimations.

Function Composition

Function composition involves creating a new function by substituting one function into another, essential for solving complex problems. In the haddock problem, we start with the weight formula based on length \( W = 0.000293 L^3 \) and then substitute the length function:\[ L(t) = 21 - 17 e^{-kt}. \]The composed function now represents weight entirely in terms of age \( t \).
This step is key because it connects separate biological measures (length and time) into a singular smoothly-behaving function\[ W(t) = 0.000293 (21 - 17 e^{-kt})^3. \]Why use function composition?

• Unification: Simplifies complex interdependent relationships into a single functional expression.
• Predictive Power: Easily analyses future scenarios by changing the input's age \( t \).
• Scalability: Enables further mathematical operations, such as calculating derivatives, without further complications.

Thus, mastering function composition allows tackling intricate system behaviors with streamlined, holistic computations.

Differentiation

Differentiation is a calculus tool used to determine the rate at which a function's value changes. Vital for finding optimal solutions, such as when the total biomass of a cohort reaches its peak. In our problem, differentiating the biomass function \( B(t) \) with respect to \( t \) helps locate this maximum.

The process involves:

• Applying the chain rule, to handle composed functions like \( W(t) \).
• Finding critical points, where the derivative \( B'(t) = 0 \).
• Evaluating these points to confirm a maximum by checking whether the function's rate of change switches from positive to negative.

Once derivative calculations reveal critical points, test within the permissible time span (up to ten years) to pinpoint where the biomass reaches its maximum size. Differentiation here ensures an accurate and efficient approach to optimize the living fish resource's mass—a critical task in resource management.