Question

Task 5 Technology may be used as an aid in this task but you must clearly explain your reasoning In this task, you will model some demographic data. A new satellite city, Springtown, has been built outside of the Brisbane metropolitan area to ease population growth in the greater Brisbane area. It was originally planned to house up to 45000 people at most. The following data is Springtown population by year since the satellite city was first started in 1998. Ongoing development is still needed into the future so population projections are required Years from 4 10 11 12 1314 15 1998 10 15.5 19.8 25.2 29 33.5 35 36.5 37.8 39 40.2 40.841.842.7 Population (000's) The Qld Govt. town planning department is modelling the population in order to plan for future population increases. They have come up with a projected population figure of 53 000 people after 20 years using a polynomial model a) Analyse the data. What assumptions have they made in this projection? Why do think they may have done this? What effect has this assumption had on the population projection? Do you agree with their projected population figure? What model would you have used? List the strengths and limitations of both models. Justify your findings mathematically b) There is another model that could be used in this case. It is called the Logistic model. This model factors in the idea that resources are not limitless and so neither is growth in things like population The equation for a LOGISTIC GROWTH FUNCTION is 772 where C, k are constants and may be found using the 1+Ce-kt data in the question ( ie. Substituting data values into the equation and manipulating the equations) and m is the maximum value y can realistically be Find the logistic equation for the population data given for Springtown manually For the various questions above include graphs as part of your written response

I just need help with question (b) - I don't know how to find c and k

Answer 1

b) The maximum population is 45000, so m= 45000. Now consider the population for t=1, t=2   With the help of these three data, C and k can be determined as follows:

b) The maximum population is 45000.so m = 45000.Now consider the population for t = 1,t = 2 with the help of these three data,C and K can be determined as follows: For t = 1, 5 = m/1 + ce^- k rightarrow 1 + ce^- k = m/5 t = 2 10 = m/1 + ce^- 2k rightarrow 1 + ce^- 2k = m/10 rightarrow 2 + 2ce^- 2k = m/5 This mean 1 + ce^- k = 2 + 2ce^- k = m/5 Therefore ce^- k = m/5 - 1 = m - 5/5 = 44995/5 (1) 2ce^- 2k = m/5 - 2 = m/5 - 10 = 44990/5 (2) Now divide (1)by(2) 1/2e^- k/e^- 2k = e^k/2 = 44995/44990 e^k = 2(44995)/44990 lne^k = k = ln 2(44995)/44990 = 0.693 using 1 + ce^- k = m/s rightarrow

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