Question

A forest region had a population of 1300 deer in the year 2000 . During the next 8 years, the deer population increased by about 60 deer per year. (a) Write a linear equation giving the deer population \(P\) in terms of the year \(t\). Let \(t=0\) represent 2000 . (b) The deer population keeps growing at this constant rate. Predict the number of deer in 2012 .

Short answer

The linear equation representing the deer population in terms of the year is \(P(t) = 60t + 1300\). The predicted deer population for 2012 is \(P(12) = 60*12 + 1300 = 2020\).

Step-by-step solution

Determine the initial condition

From the problem, we are given that in the year 2000, which we let to be represented by \(t=0\), the deer population was 1300. Hence, we have our initial condition \(P(0) = 1300\).

Set up the slope

We know that the deer population increased by about 60 deer each year. This rate of increase is the slope of our linear equation, which we represent by \(m\), and it corresponds to \(\Delta P / \Delta t\). Therefore, \(m = 60\).

Formulate the linear equation

Having the initial condition and the slope, we can now write our linear equation which has the standard form \(P(t) = mt + P(0)\). Substituting \(m = 60\) and \(P(0) = 1300\), we get the linear equation: \(P(t) = 60t + 1300\).

Predict the population for 2012

To predict the population in 2012, we substitute \(t = 12\) in our linear equation, (because 2012 is 12 years from our initial year of 2000), i.e., \(P(12) = 60*12 + 1300\). The computation gives the predicted deer population for 2012.

Key concepts

Algebraic Modelling

Algebraic modelling is a mathematical approach that involves using algebra to create equations or models that represent real-world situations. In our deer population exercise, algebraic modelling is used to convert observed population changes into a mathematical expression. The initial condition, where the deer population is 1300 at the starting year 2000, and the consistent growth rate of 60 deer per year, are pieces of information we use to construct an algebraic model.

For example, algebraic modelling lets us depict the relationship between time, represented by the variable t, and the deer population, represented by P. Our model is the linear equation P(t) = 60t + 1300, succinctly summarizing this relationship. The ability to codify such patterns into an equation is powerful because it not only provides a snapshot of the population at any given year but also allows us to predict future populations given the current trend, assuming all conditions remain constant.

Slope of a Line

The slope of a line in algebra represents the rate of change between two variables. It is calculated as the rise over the run, or in the context of our exercise, the change in population over time. Signified by the letter m, the slope is a central component in understanding how the deer population changes year over year.

Our problem gives us a slope of m = 60, indicating that for every year that passes (the run'), the population increases by 60 deer ('the rise'). A positive slope signifies that the population is growing over time. If we were to plot this on a graph, the slope would show a line ascending from left to right, where the x-axis represents time and the y-axis represents the deer population. Understanding the concept of the slope is pivotal in predicting population growth and comprehending the rate of change in many other linear relationships.

Predicting with Linear Equations

Predicting with linear equations is an invaluable tool across various disciplines, such as biology, economics, and physics. It involves using a linear equation to estimate future values based on current data trends. Our deer population scenario is a textbook example, where we extrapolate the number of deer in future years given their growth rate.

By plugging in a specific value of t (which in our case represents the number of years since 2000), we can calculate the predicted population for that year. To predict the deer population in 2012, we simply use the linear equation P(t) = 60t + 1300, with t = 12. The result of the computation, P(12), predicts the number of deer twelve years after the year 2000. This approach translates qualitative observations into quantitative forecasts, helping in planning and decision-making processes where understanding future scenarios based on current trends is crucial.