Question 3

The village population grows by a fixed number of people each year.

Step 1 — Calculate population after 6 years

We start with the initial population. People move in every year. We need to find the total increase. Then, we add it to the initial population.

$\text{Initial population} = 750$

$\text{People moving in per year} = 50$

$\text{Number of years} = 6$

$\text{Total people moved in} = 50 \times 6$

$= 300$

$\text{Population after 6 years} = 750 + 300$

$\boxed{1050 \text{ people}}$

Step 2 — Create the population table

Let's make a table. We will show population for each year. The population starts at 750. It increases by 50 each year.

| t (years) | P (population) | |---|---| | 0 | 750 | | 1 | 800 | | 2 | 850 | | 3 | 900 | | 4 | 950 | | 5 | 1000 | | 6 | 1050 | | 7 | 1100 | | 8 | 1150 | | 9 | 1200 | | 10 | 1250 |

Step 3 — Find the expression and explain linear growth

Let $P$ be the population. Let $t$ be the number of years. The initial population is 750. The population increases by 50 each year. So, after $t$ years, the increase is $50 \times t$. We add this to the initial population.

$P = \text{Initial population} + (\text{Annual increase} \times t)$

$P = 750 + (50 \times t)$

$\boxed{P = 750 + 50t}$

This represents linear growth. The population increases by a constant amount. It increases by 50 people every year. This constant rate of change defines linear growth.

Answer

(i) The population of the village after 6 years is 1050. (ii) | t (years) | P (population) | |---|---| | 0 | 750 | | 1 | 800 | | 2 | 850 | | 3 | 900 | | 4 | 950 | | 5 | 1000 | | 6 | 1050 | | 7 | 1100 | | 8 | 1150 | | 9 | 1200 | | 10 | 1250 | (iii) The expression relating $P$ and $t$ is $P = 750 + 50t$. This represents linear growth because the population increases by a constant number (50 people) every year.