Many fields of engineering require accurate population estimates. For example, t

Question

Many fields of engineering require accurate population estimates. For example, transportation engineers might find it necessary to determine separately the population growth trends of a city and adjacent suburbs. The population of the urban area is declining with time according to k,,t Pu (t) P P u min u,max while the suburban population is growing, as in kst) max PS (t) P max 1]e P0 Where P may, kur Ps max, PO and ks are empirically derived parameters. Determine the time and corresponding values of Pu Ct) and PS (t) when the suburbs are 20% larger than the city. The parameter values are u,max 75,000 0.045 yr Pumin 100,000 300,000 max PO 10,000 people 0.08 yr

Explanation / Answer

The condition given to us is

Ps(t) = Pu(t) + 20% of pu(t)

or Ps(t) = 1.2Pu(t)

Ps,max/[1+(Ps,max/P0 - 1)e^(-ks*t) ] = 1.2*(Pu,max e^(-ku*t) + Pu,min)

300000/[1+(300000/10000 - 1)e^(-.08*t) ] = 1.2*(75000 e^(-.045*t) + 100000)

300000/[1+(30 - 1)e^(-.08*t) ] = 1.2*(75000 e^(-.045*t) + 100000)

=> t = 39.6068 years

Hence the corresponding populations are :

So, Ps(t) = Ps,max/[1+(Ps,max/P0 - 1)e^(-ks*t) ]

             = 300000/[1+(300000/10000 - 1)e^(-.08*38.6068) ]

             = 135142.489 = 135143

Pu(t) = (Pu,max e^(-ku*t) + Pu,min)

       = (75000 e^(-.045*39.6068) + 100000)

       = 112618.728 = 112619

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