Question

The population of a small town was 20,000 on January 1st, 2010 and has grown at a rate of about 2% per year since then. Next year, the local high school is closing and this is likely to affect the population of the town as people move away to towns with more schools.

(a) Write down a model of the population of the town for the years 2010 to 2019. Carefully define any variables you use and state any assumptions you make.

(b) Calculate the expected size of the population on January 1st 2020.

(c) The town council wants to work out what will happen to the population once the school closes. They are considering three possible effects:

• Half the families with children of high school age will move out of town immediately. There are 1200 people in families with children of high school age in 2019.

• 10% of the population will move away each year, starting in 2020, because of decreasing opportunities for work in the town resulting from the school closure.

• 100 people who do not like children will move to the town each year once the school has closed and families start to leave. Explain how you would modify the model you wrote down in part (a) to model the population from 2020 onwards. Take account of each of the three effects. Carefully state any assumptions you make.

(d) Use dfield and your model from part (c) to estimate the size the population of the town on January 1st 2025. Include an appropriate printout from dfield to support your answer.

(e) What do you think will happen to the population of the town in the long term? Include an appropriate printout from dfield to support your answer.

Answer 1

population on January 1st, 2010 = 20,000

Let current year be y

Current population be f(y)

Answer for part (a):

Population model :

f(y) = 20,000 * (1.02) ^ (y - 2010)

where y is the year. This model is valid for 2010 <= y <= 2019

Answer for part (b):

expected size of the population on January 1st 2020 = 20000 * 1.02 ^ 10 = 24380

Answer for part (c):

New model:

Population on January 1st 2020= 24380

Now as per first effect 600 people will move out. So revised population = 24380 - 600 = 23780

f(y) =23780 * {(0.9) ^ (y - 2020)} + 100 * (y - 2020)

for 2020 <= y

The factor {(0.9) ^ (y - 2020)} is for: 10% of the population will move away each year, starting in 2020

The factor 100 * (y - 2020) is for : 100 people who do not like children will move to the town each year

Answer for part (d):

Population on January 1st 2025 = 23780 * {(0.9) ^ (2025 - 2020)} + 100 * (2015 - 2020) = 14542

Answer for part (e):  

In long run the population will keep on reducing unless a new high school opens in the town.