Question

unit 4: polynomial functions
problem 10. the population of deer in south hadley grows according to a linear function. in 2020, there were 120 deer. in 2022, there were 186 deer. this question has three parts:

• (a) find a linear function that models the population of deer in terms of years after 2020.
• (b) how many deer will there be in 2026?
• (c) in about what year will the deer population reach 240?

Explanation:

Part (a)

Step1: Define variables

Let \( t \) be the number of years after 2020, and \( P(t) \) be the population of deer. A linear function has the form \( P(t)=mt + b \), where \( m \) is the slope and \( b \) is the y - intercept.
In 2020 (\( t = 0 \)), \( P(0)=120 \), so when \( t = 0 \), \( P(0)=m\times0 + b=b \), thus \( b = 120 \).

Step2: Calculate the slope

We know two points: when \( t = 0 \) (2020), \( P = 120 \); when \( t=2022 - 2020=2 \) (2022), \( P = 186 \).
The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{186 - 120}{2-0}=\frac{66}{2}=33 \).

Step3: Write the linear function

Since \( m = 33 \) and \( b = 120 \), the linear function is \( P(t)=33t + 120 \).

Step1: Determine the value of \( t \) for 2026

2026 is \( 2026 - 2020 = 6 \) years after 2020, so \( t = 6 \).

Step2: Substitute \( t = 6 \) into the linear function

We use the function \( P(t)=33t + 120 \). Substitute \( t = 6 \): \( P(6)=33\times6+120 \).
First, calculate \( 33\times6 = 198 \), then \( 198+120=318 \).

Step1: Set up the equation

We want to find \( t \) when \( P(t)=240 \). Using the function \( P(t)=33t + 120 \), we set \( 33t+120 = 240 \).

Step2: Solve for \( t \)

Subtract 120 from both sides: \( 33t=240 - 120=120 \).
Then, divide both sides by 33: \( t=\frac{120}{33}=\frac{40}{11}\approx3.64 \).

Step3: Find the year

Since \( t \) is the number of years after 2020, the year is \( 2020+\lfloor t
floor + 1\) (we round up because we are looking for the year when the population reaches 240). \( 2020 + 4=2024 \) (since \( t\approx3.64 \), we need about 4 years after 2020).

Answer:

\( P(t)=33t + 120 \) (where \( t \) is the number of years after 2020)

Part (b)