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.
Step-by-step solution
Idea: The population starts at 750 and gains a fixed 50 people a year (taking only this movement into account), so after t years it has gained 50t.
(i) Find the population of the village after 6 years.
- Increase in 6 years = 50 × 6 = 300.½ mark
- Population = 750 + 300 = 1050.½ mark
(ii) Make a table of values for t varying from 0 to 10 years and show how the population, P, increases every year.
- Add 50 for each year:1 mark
Year, t Population, P 0 750 1 800 2 850 3 900 4 950 5 1000 6 1050 7 1100 8 1150 9 1200 10 1250 - The population increases by the same 50 every year.½ mark
(iii) Find an expression that relates P and t, and explain why it represents linear growth.
- P = 750 + 50t1 mark
- It is linear in t with a positive coefficient 50: the population increases by a constant 50 over each equal interval of one year. So it represents linear growth.½ mark
Check: P(10) = 750 + 500 = 1250, the last entry of the table ✓.
Answer to write in the exam
(i)
Increase in 6 years = 50 × 6 = 300
Population = 750 + 300
∴ Population after 6 years = 1050
(ii)
t: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
P: 750, 800, 850, 900, 950, 1000, 1050, 1100, 1150, 1200, 1250
∴ P increases by 50 every year
(iii)
P = 750 + 50t
P increases by a constant 50 in each year (equal intervals)
∴ It represents linear growth
Common mistakes that cost marks
- Writing P = 800t (adding 750 + 50 and multiplying by t).
- Starting the table at 800 for t = 0. At t = 0 the population is the initial 750.
- Treating the growth as a percentage. Here a fixed number, 50, is added each year.
How this can come in the exam
A town has 2400 people and gains 120 people every year. The population after t years is
- 2400 + 120t
- 2520t
- 2400t + 120
- 120 − 2400t
Show answer
(A) 2400 + 120t
Start at 2400 and add 120 for each of the t years.
For the village (P = 750 + 50t), after how many years will the population reach 1500?
Show answer
750 + 50t = 1500 (1 mark) ⇒ 50t = 750 ⇒ t = 15 years (1 mark).Try one yourself
A school has 420 students and admits 35 extra students every year. Write S in terms of t and find S after 8 years.
Show answer
S = 420 + 35t; after 8 years 700 students.
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