Question

The state game commission introduces 30 elk into a new state park. The population $N$ of the herd is modeled by $N=[10(3+4t)]/(1+0.1t)$ where $t$ is the time in years. (a) Find the size of the herd after 5,10 , and 25 years. (b) According to this model, what is the limiting size of the herd as time progresses?

Short answer

The sizes of the herd after 5, 10, and 25 years are 60, 80, and 85 elks respectively. The limiting size of the herd as time progresses is around 40 elks.

Step-by-step solution

Calculate the size of the herd after 5, 10 and 25 years

Substitute $t=5$, $t=10$, and $t=25$ into the function $N(t)=\frac{10(3+4t)}{1+0.1t}$ and calculate the values respectively. This will give the size of the herd after 5, 10 and 25 years.

Calculation

After substituting $t=5$ into the function, we get $N(5)=\frac{10(3+4\ast 5)}{1+0.1\ast 5}=60$. Similarly, after substituting $t=10$, we get $N(10)=\frac{10(3+4\ast 10)}{1+0.1\ast 10}=80$, and after substituting $t=25$ we get $N(25)=\frac{10(3+4\ast 25)}{1+0.1\ast 25}=85$.

Determine the limiting size of the herd as time progresses

To find the limiting size of the herd, you need to take the limit of the function $N(t)$ as $t$ approaches infinity. According to the laws of limits, when you have a rational function (a fraction in which both the numerator and denominator are polynomials), the limit as $t$ approaches infinity is determined by the highest degrees of $t$ in the numerator and the denominator. In our case, the highest degree in both the numerator and denominator is 1. Since the coefficients are the same (4 in the numerator and 0.1 in the denominator), the limit as $t$ approaches infinity will be the ratio of these coefficients, or $\frac{4}{0.1}=40$.

Interpret the results

Therefore, the sizes of the elk herd after 5, 10, and 25 years are 60, 80, and 85 respectively. The limiting size of the herd as time progresses is 40. This means that, according to the model, the size of the herd tends to stabilize around 40 as time progresses.

Key concepts

Limits in Calculus

Understanding the concept of limits in calculus is fundamental when dealing with population modeling. Limits help us understand what happens to a function as the input values approach a certain point. For instance, in \textbf{population modeling}, to predict the eventual number of individuals in a population, we use limits to calculate the 'carrying capacity' which is the maximum population size that the environment can sustain.

In the case of the elk herd modeled by the equation $N(t)=\frac{10(3+4t)}{1+0.1t}$, we are interested in finding out what happens to the population as time goes on indefinitely, symbolically represented as $t$ approaches infinity. The term 'limit' signifies that we are not concerned with the population size at any one specific time, but rather the trend it follows as time becomes extremely large. Mathematically, we write this as $\underset{t\to \mathrm{\infty }}{lim}N(t)$. When this calculus method indicates a finite value, it suggests that the population will stabilize at that size.

When applying limits to rational functions like our model, if the highest power of $t$ is the same in both the numerator and the denominator, the limit value is the ratio of their coefficients. This is why the herd's limiting size is calculated to be 40, a critical insight for ecologists and park managers planning for sustainable wildlife management.

Rational Function Behavior

In algebra, understanding the behavior of rational functions is incredibly important for interpreting models such as the one used for the elk population. A rational function is characterized by an equation that takes the form of a fraction where both the numerator and denominator are polynomial expressions.

With the elk herd, our rational function is $N(t)=\frac{10(3+4t)}{1+0.1t}$, showcasing how the population changes over time. The key to predicting population trends lies in scrutinizing how the function behaves as $t$ increases. One important feature to notice is that as $t$ becomes very large, the $4t$ term in the numerator and the $0.1t$ term in the denominator dominate the other terms, essentially determining the function's long-term behavior.

This is exemplified in our model calculations for the population sizes at various times. Initially, for smaller values of $t$, other terms play a role, yet over the long term, as $t$ grows, those terms become negligible, and the ratio of the leading coefficients, 4 to 0.1, provides the horizontal asymptote value of 40. This horizontal asymptote indicates where the population settles as time progresses to infinity, a key insight when managing wildlife populations to prevent overpopulation or extinction.

Algebraic Modeling

The process of representing real-world situations with mathematical equations is known as algebraic modeling. It involves translating the patterns or behaviors we observe into algebraic expressions or functions. In our exercise, the state game commission wants to predict and manage the elk population in a new state park, so it uses an algebraic model $N(t)=\frac{10(3+4t)}{1+0.1t}$ to serve this purpose.

The algebraic model relies on assumptions and data about elk population dynamics. The constants and variables in our function each have a significance: the '10' adjusts the scale of the model to fit the real-world population size, '3+4t' captures the growth aspect of the population over time, while '1+0.1t' may represent factors limiting growth, such as environmental resistance.

Algebraic modeling enables scientists and researchers to analyze trends, make predictions, and better understand underlying patterns within the data. When the model's predictions are compared to actual data, they can validate the model or suggest areas for refinement. In effect, algebraic modeling serves as a bridge between mathematical theory and real-life applications, helping us to interpret and navigate the complexities of natural phenomena like wildlife population dynamics.