The population P of elk on a refuge is changing at a rate of (d P)/(d t)=(25.0)/(1.00+0.100 t),

step 1 In the problem, we have dp upon d t, that is 25 upon 1 plus 0 .1 t. So here we assume 1 plus 0 .

The population P of elk on a refuge is changing at a rate of...

The population $P$ of elk on a refuge is changing at a rate of $\frac{d P}{d t}=\frac{25.0}{1.00+0.100 t},$ where $t$ is the time in years. If the original population (when $t=0$ ) was 125 elk, find the population 5 years later.

Key Concepts

Differential Equation Modeling

Differential equations are used to represent systems in which a quantity changes over time. In the context of population dynamics, the rate of change of the population is described by a differential equation, which models how the population increases or decreases due to various factors. This approach provides a mathematical framework to describe real-life phenomena such as growth or decay processes.

Initial Value Problem

An initial value problem (IVP) specifies the value of the function at a certain point, which in turn allows us to determine the constant of integration when solving the differential equation. In problems like the population growth example, knowing the initial population helps in finding the particular solution that fits the situation, ensuring the model accurately represents the system's initial condition.

Integration by Substitution

Integration by substitution is a technique used to simplify the integration process by changing variables. When a differential equation involves a function that is difficult to integrate in its current form, this method can be applied to transform the integrand into a simpler form. In the context of the problem, the substitution method is used to integrate a rational function, which ultimately leads to a logarithmic expression that is essential for finding the solution to the population model.

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