Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

The wolf population \(P\) in a certain state has been growing at a rate proportional to the cube root of the population size. The population was estimated at 1000 in 1980 and at 1700 in 1990 . (a) Write the differential equation for \(P\) at time \(t\) with the two corresponding conditions. (b) Solve the differential equation. (c) When will the wolf population reach 4000 ?

Step-by-Step Solution

Answer

The wolf population will reach 4000 approximately 17.8 years after 1980.

Step 1: Understanding the Differential Equation

The growth rate of the population is proportional to the cube root of the population size. Thus, the differential equation can be written as \( \frac{dP}{dt} = k \cdot P^{1/3} \), where \( k \) is the proportionality constant.

Step 2: Initial Conditions

We know that the population \( P \) was 1000 in 1980 and 1700 in 1990. These can be written as initial conditions: \( P(0) = 1000 \) and \( P(10) = 1700 \).

Step 3: Solve the Differential Equation

To solve \( \frac{dP}{dt} = k \cdot P^{1/3} \), we separate variables: \( \frac{dP}{P^{1/3}} = k \, dt \). Integrating both sides gives us \( \int P^{-1/3} \, dP = \int k \, dt \). This results in \( 3P^{2/3} = kt + C \), where \( C \) is the integration constant.

Step 4: Apply Initial Condition to Find C

Using the initial condition \( P(0) = 1000 \), substitute into the equation \( 3 \cdot (1000)^{2/3} = k \cdot 0 + C \). Evaluating, we find \( C = 3 \cdot 1000^{2/3} = 3 \cdot 100 \cdot 10^{2/3} = 300 \).

Step 5: Determine Constant k

Using the other condition \( P(10) = 1700 \), substitute into \( 3 \cdot (1700)^{2/3} = 10k + 300 \) and solve for \( k \). This leads to \( k = \frac{3 \cdot 1700^{2/3} - 300}{10} \).

Step 6: General Solution

With \( k \) and \( C \) found, the general form is \( P = \left( \frac{kt + 300}{3} \right)^{3/2} \). Substitute the value of \( k \) to complete the equation.

Step 7: Find When Population Reaches 4000

Set \( P = 4000 \) and solve \( \left( \frac{kt + 300}{3} \right)^{3/2} = 4000 \) to find \( t \). After calculating, we solve \( kt + 300 = 3 \cdot 4000^{2/3} \) for \( t \).

Key Concepts

Population Growth

Population growth in the context of differential equations can take various forms. In our example, we're dealing with a situation where the wolf population grows at a rate proportional to the cube root of its size. This is an interesting model because it suggests that as the population increases, the rate of growth changes in a less than straightforward manner.

Understanding how populations adapt and modify their growth rates helps in predicting long-term trends. These models can be especially useful in ecological management and conservation efforts. To sum up the core idea:

• The growth rate depends not directly on the population size, but on its root value.
• This relationship complicates the prediction of future population sizes.

Initial Conditions

Initial conditions are vital for solving any differential equation that describes real-world phenomena. In our exercise, these conditions are given by the population sizes at two different times: 1000 in 1980, and 1700 in 1990. These known values are used to determine unspecified constants in our solution.

By applying initial conditions, we can:

• Determine the specific integration constant necessary for the solution.
• Validate the model by ensuring that it aligns with past observed data points.

This aspect of solving differential equations ensures that the solutions are relevant and applicable to real-world situations.

Separation of Variables

Separation of variables is a core technique in solving differential equations. It involves rearranging an equation so that each variable and its derivative are isolated on opposite sides of the equation. In this exercise, we used this technique by manipulating the equation:

\( \frac{dP}{dt} = k \cdot P^{1/3} \)

This is then rewritten as:

\( \frac{dP}{P^{1/3}} = k \, dt \)

The next step is to integrate both sides: \( \int P^{-1/3} \, dP = \int k \, dt \).

Separation of variables is particularly effective when equations can be expressed in a way that integrates easily, providing a straightforward path to a solution. By separating variables, you make it easier to think about how changes in one variable affect another.

Proportionality Constant

The proportionality constant, denoted as \(k\) in our differential equation, is crucial for defining the exact rate at which the wolf population grows. Without estimating this constant accurately, predicting future population sizes would be difficult. It represents the efficiency or intensity of population growth in response to the existing population.

In this exercise:

• We used two initial conditions to help solve for \(k\).
• Using values from known population data, we calculated \(k\) through substitution and solving algebraic equations.

Understanding \(k\) paints a fuller picture of how rapid changes in the environment or resources can influence population growth rates. It's a pivotal component in modeling dynamic systems such as wildlife populations.