Question

in 1991, the moose population in a park was measured to be 4210. by 1997, the population was measured again to be 5170. 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 2004?

Explanation:

⚡ Using what you learned: Creating Equations to Model Relationships

Step 1: Identify the data points

Let \( t \) represent the number of years since 1990.

• For the year 1991:

$$ t = 1991 - 1990 = 1 $$

The population is \( P(1) = 4210 \). This gives the point \( (1, 4210) \).

• For the year 1997:

$$ t = 1997 - 1990 = 7 $$

The population is \( P(7) = 5170 \). This gives the point \( (7, 5170) \).

Step 2: Find the rate of change (slope)

Calculate the slope \( m \):

$$ m = \frac{5170 - 4210}{7 - 1} = \frac{960}{6} = 160 $$

Step 3: Find the linear formula

Using the point-slope form with \( (1, 4210) \):

$$ P(t) - 4210 = 160(t - 1) $$

$$ P(t) = 160t - 160 + 4210 $$

$$ P(t) = 160t + 4050 $$

Step 4: Predict the population in 2004

For the year 2004:

$$ t = 2004 - 1990 = 14 $$

Substitute \( t = 14 \) into the formula:

$$ P(14) = 160(14) + 4050 $$

$$ P(14) = 2240 + 4050 = 6290 $$

Answer:

A.) \( P(t) = 160t + 4050 \)

B.) \( 6290 \)