In population modeling, integration provides a systematic way to determine accumulated quantities from known rates of change. One such application arises in ecology, where the total weight of a fish population in a body of water is referred to as its biomass. When the rate of growth of this biomass is known as a function of time, calculus can be used to determine the total biomass at a future date.
Growth Rate and Biomass Function
Let the growth rate of the fish population be represented by a function G(t), where t denotes time measured in years and G(t) represents the rate of change of biomass in kilograms per year. The biomass itself is represented by a function B(t). Since the growth rate describes how the biomass changes over time, the biomass function is obtained by integrating the growth rate function with respect to time.
Use of the Substitution Method
To evaluate the integral defining the biomass function, the substitution method is employed. This involves selecting an appropriate substitution variable u and its differential du to simplify the integrand. After performing the substitution, the integral is rewritten in terms of u, making it easier to evaluate. Once the integration is complete, the expression is converted back to the original variable t, resulting in a general form of the biomass function that includes an unknown constant of integration.
Applying Initial Conditions
The initial biomass of the fish population in the year 2000 is given and serves as an initial condition. By substituting the corresponding time value and biomass into the general biomass function, the constant of integration can be determined. This step ensures that the model accurately reflects the known state of the system at the starting time. After finding this constant, the biomass function is rewritten in its final form.
Determining Biomass in 2005
To find the biomass of the fish population in the year 2005, the time variable is set to five years after 2000. Substituting this value into the finalized biomass function and simplifying yields the total biomass of the fish population in 2005.
In a water body, the total weight of the fish population is known as the biomass of fish. The rate of growth of a fish population can be modeled by the function G(t), where t is the number of years and G(t) is the growth rate measured in kilograms per year.
The initial biomass for the year 2000 is given.
The goal is to find the biomass of the fish population in the year 2005.
To find this, the rate function is integrated to find the biomass function B(t).
The substitution method is used to solve the integral by taking u and its derivative du.
The resulting integral in u is simplified, and the expression is then written back to get B(t) with an unknown constant c.
The known quantities are taken as the initial conditions. After substituting these values and solving, the unknown constant c is found. Then, substituting this c into the equation, B(t) is rewritten.
To find the biomass in the year 2005, t equals 5 is substituted into this biomass equation.
After simplifying the equation, the biomass function is found, showing the total biomass of the fish population in 2005.
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