Problem 71
Question
Fish Growth von Bertalanffy's equation is used to model the growth of fish. The length of the fish, \(L(t)\), grows at a rate that depends on its current length \(L(t)\) (that is, big and small fish grow at different \mathrm{\\{} r a t e s ) . ~ The growth of a fish is modeled using the differential equation: $$ \frac{d L}{d t}=k\left(L_{\infty}-L\right) $$ where \(k\) and \(L_{\infty}\) are both positive constants. To solve the equation, we will learn in Chapter 8 that it is necessary to calculate the following integral: $$ t=\int \frac{d L}{k\left(L_{\infty}-L\right)} $$ Assume \(k=L_{\infty}=1 ;\) then evaluate the integral. $$ t=\int \frac{d L}{1-L} $$ Your answer will contain an unknown constant of integration.
Step-by-Step Solution
Verified Answer
The integral evaluates to \(t = -\ln|1-L| + C\), where \(C\) is a constant of integration.
1Step 1: Substitute constants into the integral
Given that \(k = L_{\infty} = 1\), the integral simplifies from \(t = \int \frac{dL}{k(L_{\infty} - L)}\) to \(t = \int \frac{dL}{1 - L}\). This will be the new expression we need to evaluate.
2Step 2: Recognize the integral form
The integral \(\int \frac{dL}{1 - L}\) is a standard form that can be solved using a simple substitution or recognizing it as an integral of a rational function. The antiderivative of \(\frac{1}{1-L}\) is \(-\ln|1-L|\).
3Step 3: Evaluate the integral
Use the known antiderivative: \(t = -\ln|1-L| + C\), where \(C\) is the constant of integration. This provides the general solution to the integral.
Key Concepts
Integral CalculusFish Growth Modelsvon Bertalanffy's Equation
Integral Calculus
Integral Calculus is an essential branch of mathematics that focuses on the study of integrals and their properties. In simpler terms, it deals with the accumulation of quantities, such as area under a curve. This is useful for calculating the total growth of a quantity over time or space.
In our exercise, we were asked to compute an integral of the form: \[ \int \frac{dL}{1-L} \] This represents the accumulation of the growth rate of a fish over time. We relied on the concept of antiderivatives to perform this calculation, which involves finding a function that reverses the differentiation process. The antiderivative of \( \frac{1}{1-L} \) is \(-\ln|1-L|\).
Integrating rational functions like this often involves recognizing standard integral forms or using substitution methods. This integral gives us insights into how the length of a fish changes over time, linking the length directly to time via the natural logarithm.
In our exercise, we were asked to compute an integral of the form: \[ \int \frac{dL}{1-L} \] This represents the accumulation of the growth rate of a fish over time. We relied on the concept of antiderivatives to perform this calculation, which involves finding a function that reverses the differentiation process. The antiderivative of \( \frac{1}{1-L} \) is \(-\ln|1-L|\).
Integrating rational functions like this often involves recognizing standard integral forms or using substitution methods. This integral gives us insights into how the length of a fish changes over time, linking the length directly to time via the natural logarithm.
Fish Growth Models
Understanding the growth of fish is crucial for fields like ecology and fisheries management. Fish Growth Models are mathematical descriptions that mimic how fish grow over time. These models help predict changes in fish populations and assist in sustainable fisheries management.
Fish growth can be influenced by various factors such as food availability, environmental conditions, and genetics. That's why creating accurate models is important. In the given problem, the growth of the fish length \( L(t) \) is determined using the differential equation: \[ \frac{dL}{dt} = k(L_\infty - L) \] Here, \( L_\infty \) represents the maximum achievable length of the fish, and \( k \) is the growth rate constant.
This equation implies an interesting relationship: as a fish grows larger, it grows more slowly, approaching but never quite reaching \( L_\infty \). Such models are vital in understanding biological growth processes.
Fish growth can be influenced by various factors such as food availability, environmental conditions, and genetics. That's why creating accurate models is important. In the given problem, the growth of the fish length \( L(t) \) is determined using the differential equation: \[ \frac{dL}{dt} = k(L_\infty - L) \] Here, \( L_\infty \) represents the maximum achievable length of the fish, and \( k \) is the growth rate constant.
This equation implies an interesting relationship: as a fish grows larger, it grows more slowly, approaching but never quite reaching \( L_\infty \). Such models are vital in understanding biological growth processes.
von Bertalanffy's Equation
Von Bertalanffy's Equation is a specific type of fish growth model. It is represented by the differential equation:\[ \frac{dL}{dt} = k(L_\infty - L) \] This equation captures the concept of diminishing returns in growth. The equation implies that the growth rate of a fish slows down as it nears its maximum size, \( L_\infty \).
Key points about von Bertalanffy's Equation:
Key points about von Bertalanffy's Equation:
- It models growth in an asymptotic manner, meaning growth decreases as size increases.
- The parameter \( k \) determines how quickly the fish approaches its maximum size.
- Useful for modeling not only fish, but also other living organisms that grow in a similar pattern.
Other exercises in this chapter
Problem 70
The Gompertz equation is used to model the growth of a tumor. We will study it in Chapter \(8 .\) In this model the number of cells \(N(t)\) in a tumorgrows ove
View solution Problem 70
Evaluate each integral. $$ \int \frac{2 x-1}{(x+4)(x+1)} d x $$
View solution Problem 71
Evaluate each integral. $$ \int \frac{1}{x^{2}-9} d x $$
View solution Problem 72
Fish Growth von Bertalanffy's equation is used to model the growth of fish. The length of the fish, \(L(t)\), grows at a rate that depends on its current length
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