Modeling with Linear Functions

Question

In 1994, the moose population in a park was measured to be 4250. By 1996, the population was measured again to be 3990. If the population continues to change linearly:

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

   What does your model predict the moose population to be in 2002? 

Answer 1

It is given that this population changes linearly. Thus, we can use slope-intercept formula to find a formula P(t).

If we let (t, P(t)) = (year, moose population), our two points are (4,4250) and (6, 3990) because we are measuring the number of years after 1990.

First, let's find slope. m = (P(t)2 - P(t)1) / (t2 - t1) = (3990 - 4250) / (6 - 4) = -260 / 2 = -130

Next, let's plug one of the points and the slope into slope-intercept form to find the intercept, b

P(t) = mt + b

where m = slope and b = intercept

We are going to plug in the point (4, 4250)

4250 = (-130) * (4) + b

4250 = -520 + b

add 520 to both sides

4770 = b

Now, we have our answer to part one: P(t) = -130t + 4770

Part two asks us to predict what the moose population will be in 2002. First, we need to find our t. 2002 - 1990 = 12

Because 2002 is 12 years after 1990, t = 12

Now, we plug into the formula above:

P(12) = -130 * 12 + 4770

P(12) = 3210 is our answer for part two