Question

In 1994, the moose population in a park was measured to be 4930. By 1999, the population was measured again to be 6180. If the population continues to change linearly: A.) Find a formula for the moose population, P, in terms of t, the years since 1990. P(t) B.) What does your model predict the moose population to be in 2006?

Answer 1

We define the following variables for our problem:

P = population of mooses

t = number of years since 1990

m = growth ratio

In terms of the variables that we defined above and the fact that the population of moose in terms of the year is linear, we have the following equation:

`P(t)=m\cdot t+P_0`

Now, we use the data of the problem:

1) In 1994 the moose population was 4930, so we have:

`\begin{gathered} t=1994-1990=4 \\ P(4)=4930 \end{gathered}`

2) In 1999 the moose population was 6180, so we have:

`\begin{gathered} t=1999-1990=9 \\ P(9)=6180 \end{gathered}`

Now, using the data above and the equation for P(t) we construct the following system of equations:

`\begin{gathered} P(4)=m\cdot4+P_0=4930 \\ P(9)=m\cdot9+P_0=6180 \end{gathered}`

We solve the system of equations.

First, we solve the equations for P0:

`\begin{gathered} P_0=4930-4m \\ P_0=6180-9m \end{gathered}`

Now, because the right-hand-side of both equations is equal to P0, we equal them and then we solve for the variable m:

`\begin{gathered} 4930-4m=6180-9m \\ 9m-4m=6180-4930 \\ 5m=1250 \\ m=250 \end{gathered}`

Finally, we replace the value of m in one of the equations of P0 and solve for it:

`\begin{gathered} P_0=4930-4\cdot m \\ P_0=4930-4\cdot250 \\ P_0=3930 \end{gathered}`

A) The formula for the moose population, P, in terms of t, the years since 1990 is:

`P(t)=250t+3930`

B) We want to know the value of the moose population in 2006.

First, we compute the value of t:

`t=2006-1990=16`

Now, we replace the value of t in the equation of P(t) above:

`P(6)=250\cdot16+3930=7930`

Answer: 7930