Question

3. (13 marks) 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 decreas- ing 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 pop- ulation from 2020 onwards. Take account of each of the three effects. Carefully state any assumptions you make. (d) Une 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. (@) What do you think will happen to the population of the town in the long term? Include an appropriate printout from yfield to support your answer. 4. (20 marks This question is about the linear system

Answer 1

GIVEN: POPULATION ON 01.01.2040 = 20,000 GROWTH RATE = 2% MODEL FOR POPULATION TILL 2019 (a) P = 20000 (12) P=20000 (140.02) POR P= 20000 crost OR BERDOA Dost (8) EXPECTED Size Of POPULATION ON 01.01.2020 = 20000 (1+0.02) = 20000 * 1218994 = 24380 POPULATION EXPECTED ON 01.01.2020 - 24,380 OUT OF TOWN (c)- Half of 1200 (1. e. = 1260=600) WILL MOVE IMMEDIATELY LL MOVE AWAY e REDUCE EACH YEAR 100 PEOPLE WILL COME "TO. TOWN GACH YEAR WHO DOES NOT LIKE CHILDREN MODEL FOR POPULATION FROM 2020 ONWARDS WILL BE: D

P= (24380-600) (1-10 + 100€ P= 23780 (1-0.46 + +loot p=23780 (o.glt toot THEREFORE POPULATION MODEL FROM 2020 WILL BE on 93780 (0.9) toot 01.01.2025 (d) POPULATION ON p=2378060.9)” +100x5 P= 2378060.935 +500 P = 14041.85 +500 R=445 44 88 Pa 14541.85 HENCE POPULATION ON 01.01.2025 WILL BE 14541 ANS.