Moose Population Formula

Question

In 1994, the moose population in a park was measured to be 4000. By 1998, the population was measured again to be 3600. If the population continues to change linearly:

Find a formula for the moose population, P, in terms of t, the years since 1990.

P = ___________

What does your model predict the moose population to be in 2008?

Answer 1

Bottom line: We use the equation of a line, y=mx+b, to create a linear model and use the two data points given to calculate the slope m=(y₂-y₁)/(x₂-x₁), and then use one of the points to calculate our intercept (b value). This gives us the linear model, we then plug-in the date given to answer the second part of the question. Detailed work is shown below.

The key to starting this problem is the term "linearly", it tells us that we want to create a linear model, of the form y=mx+b, based on the data we are given.

Definition of our linear model" y=mx+b

1. y is our independent variable, aka the variable that we want to predict. In this problem that is the moose population
2. m is the slope, or rate of change, of our function. This value is not given to us in the problem statement
3. x is our dependent variable, aka the variable we are using to make a prediction about the value of y. In this problem it is defined as the number of years since 1990
4. b is our intercept, aka the value of y if our value of x is zero. This value is not given to us in the problem

Based on the above definition we can write our linear model as P=mt+b and we can see from this that we need to calculate the value for our slope (m), and the value for our intercept (b).

The slope is defined as the rate of change, which we can calculate using any two points on our line using the equation m=(y₂-y₁)/(x₂-x₁) where (x₁, y₁) and (x₂, y₂) are any two points on our line. The problem gives us two points in the first two sentences of the problem, aka a population of 4000 in 1994 and a population of 3600 in 1998. Note we need to convert the years to t values since t is defined as the number of years since 1990. After that conversion our two points are (4,4000) and (8,3600). We substitute into our equation for the slope to get m=(3600-4000)/(8-4)=-100

Now our linear model is P=-100t+b. We can then use one of our points in order to calculate the value of our intercept (b). Substituting in the value from our first point gives us 4000=-100(4)+b. Solving for b we add 400 to both sides, resulting in b=4400. We then substitute our value of b back into the model

At this point we have answered the first question. P=-100t+4400

To answer the second question we use the date given (2008) to calculate a value for t of 18, we then substitute into our model. P=-100 (18)+4400=2600

Answer 2