The initial population of a village is 750. Every year, 50 people move from a nearby city to the village. (i) Find the population of the village after 6 years. (ii) Make a table of values for varying from 0 to 10 years and show how the population, , increases every year. (iii) Find an expression that relates and , and explain why it represents linear growth.

Understand the Question Initial Population: The village starts with people at . Annual Increase: Every year, a constant number of people move in. Growth Model: Since the population increases by a fixed amount each year, the growth is linear, given by . Part (i) Find the population of the village after 6 years. Step 1 · Calculate Population After 6 Years Given Answer > (i) Part (ii) Make a table of values for varying from 0 to 10 years and show how the population, , increases every year. Step 1 · Construct the Table of Values Starting with at and adding for each subsequent year: Answer > (ii) Part (iii) Find an expression that relates and , and explain why it represents linear growth. Step 1 · Formulate Expression and Explain Growth Let be the population after years. This represents linear growth because the population increases by a constant rate of change ( people per year). Answer > (iii) . This represents linear growth because the population increases by a constant number ( people) every year. Common Mistakes Omitting Initial Population: Writing instead of by forgetting to add the baseline initial population. Linear vs. Exponential Growth: Confusing linear growth (constant addition of a fixed number, ) with exponential growth (multiplication by a fixed percentage/factor each year).

Introduction to Linear Polynomials | Exercise 2.4

Question 3

The initial population of a village is 750. Every year, 50 people move from a nearby city to the village.

(i) Find the population of the village after 6 years.

(ii) Make a table of values for $t$ varying from 0 to 10 years and show how the population, $P$ , increases every year.

(iii) Find an expression that relates $P$ and $t$ , and explain why it represents linear growth.

Solution

Understand the Question

Initial Population: The village starts with $750$ people at $t = 0$ .
Annual Increase: Every year, a constant number of $50$ people move in.
Growth Model: Since the population increases by a fixed amount each year, the growth is linear, given by $\text{Population} = \text{Initial Population} + (\text{Rate} \times \text{Time})$ .

(i) Find the population of the village after 6 years.

Step 1 · Calculate Population After 6 Years

Given $\text{Initial population} = 750$ $\text{People moving in per year} = 50$ $\text{Number of years} = 6$

\begin{aligned} \text{Total people moved in} &= 50 \times 6 \\ &= 300 \end{aligned}

\begin{aligned} \text{Population after 6 years} &= 750 + 300 \\ &= 1050 \end{aligned}

Answer

(i) $1050 \text{ people}$

(ii) Make a table of values for $t$ varying from 0 to 10 years and show how the population, $P$ , increases every year.

Step 1 · Construct the Table of Values

Starting with $750$ at $t = 0$ and adding $50$ for each subsequent year:

\begin{array}{|c|c|} \hline t \text{ (years)} & P \text{ (population)} \\ \hline 0 & 750 \\ \hline 1 & 800 \\ \hline 2 & 850 \\ \hline 3 & 900 \\ \hline 4 & 950 \\ \hline 5 & 1000 \\ \hline 6 & 1050 \\ \hline 7 & 1100 \\ \hline 8 & 1150 \\ \hline 9 & 1200 \\ \hline 10 & 1250 \\ \hline \end{array}

Answer

(ii) $$ \begin{array}{|c|c|} \hline t \text{ (years)} & P \text{ (population)} \ \hline 0 & 750 \ \hline 1 & 800 \ \hline 2 & 850 \ \hline 3 & 900 \ \hline 4 & 950 \ \hline 5 & 1000 \ \hline 6 & 1050 \ \hline 7 & 1100 \ \hline 8 & 1150 \ \hline 9 & 1200 \ \hline 10 & 1250 \ \hline \end{array}

(iii) Find an expression that relates $P$ and $t$ , and explain why it represents linear growth.

Step 1 · Formulate Expression and Explain Growth

Let $P$ be the population after $t$ years.

\begin{aligned} P &= \text{Initial population} + (\text{Annual increase} \times t) \\[0.6em] &= 750 + (50 \times t) \\[0.6em] &= 750 + 50t \end{aligned}

This represents linear growth because the population increases by a constant rate of change ( $50$ people per year).

Answer

(iii) $P = 750 + 50t$ . This represents linear growth because the population increases by a constant number ( $50$ people) every year.

Common Mistakes

Omitting Initial Population: Writing $P = 50t$ instead of $P = 750 + 50t$ by forgetting to add the baseline initial population.
Linear vs. Exponential Growth: Confusing linear growth (constant addition of a fixed number, $+50$ ) with exponential growth (multiplication by a fixed percentage/factor each year).

Question 3

More questions in Exercise 2.4