The city of Lincoln had a population of 37 thousand in 1900. The populations is

Question

The city of Lincoln had a population of 37 thousand in 1900. The populations is 277 thousand in 2016. Write a formula for P(t), the population of Lincoln in thousands of people, as a function of t. In this function t is the number of years since 1900, that is, 1900 should be taken as t = 0 if: The population grew by the same number of people each year. The population grew by the same percent each year. Assuming the population of Lincoln was growing exponentially, find the continuous growth rate of the population.

Explanation / Answer

p(t) is population function

in 1900 ==> t = 0 ; p(0) = 37

in 2016 ==> t = 116 ; p(116) = 277

a) population grew by same number of people

==> the rate of population growth is constant ==> dp/dt = k

Integrating on both sides with respect to t

==> (dp/dt) dt = k dt

==> p(t) = kt + c

p(0) = 37 ==> 37 = k(0) + c

==> c = 37

p(116) = 277

==> 277 = k(116) + 37

==> 116k = 277 - 37

==> 116k = 240

==> k = 240/116

==> k = 60/29

Hence p(t) = (60/29)t + 37

b) population grew by same percent every year ==> p(t) = c(1 + r)t

==> p(0) = c = 37

p(116) = 37(1 + r)116

==> 277 = 37(1 + r)116

==> (1 + r)116 = 277/37

==> ln (1 + r)116 = ln (277/37)

==> 116 ln (1 + r) = ln (277/37)

==> ln (1 + r) = (1/116) ln (277/37)

==> ln (1 + r) = 0.01735

==> 1 + r = e0.01735

==> 1 + r = 1.01751

==> p(t) = 37(1.01751)t

c) population growth is exponential ==> dp/dt = kp

==> dp /p = k dt

integrating on both sides

==> dp /p = k dt

==> ln p = kt + c

==> p = ekt + c

==> p = ekt ec

==> p(t) = Cekt

p(0) = 37 ==> 37 = Cek(0)

==> C = 37

p(116) = 37e116k

==> 277 = 37000e116k

==> e116k = 277/37

==> ln e116k = ln (277/37)

==> 116k = ln (277/37)

==> k = (1/116) ln (277/37)

==> k = 0.01735

==> p(t) = 37e0.01735t

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