The population of a town a town after t weeks is given by p(t)=1200(2^-t). a) what is the initial population of the town? b) how many people are there after 1 week? c) what is the rate of change of people after 1 week?

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Answer 1 · 2018-04-22T00:36:46

a) #1200# b) #600# #c) -600ln2approx-415.99#

a) The initial population of the town is simply #p(t)# evaluated at the smallest possible value of #t#, that is, #t=0:#

#p(0)=1200(2^0)=1200#

The initial population is #1200# people.

b) The amount of people in the town after #1# week is #p(t)# evaluated at #t=1:#

#p(1)=1200(2^-1)=1200/2=600#

There are #600# people after a week.

c) The rate of change of people after #1# week will be the first derivative of #p(t)# evaluated at #t=1#, IE, #p'(1).#

To differentiate, let's rewrite a little, recalling that #x=e^lnx#:

#p(t)=1200(e^ln(2^-t))#

#ln(a^b)=blna,# so #ln(2^-t)=-tln2#:

#p(t)=1200(e^(-tln2))#

Differentiate, using the Chain Rule:

#p'(t)=1200(e^(-tln2))*d/dt(-tln2)#

#d/dt(-tln2)=-ln2,# so:

#p'(t)=-1200(ln2)(e^(-tln2))#

Recalling that #e^(-tln2)=e^(ln2^-t)=2^-t#,

#p'(t)=-1200(ln2)(2^-t)#

#p'(1)=-1200(ln2)(2^-1)=-600ln2#

The rate of change after a week is #-600ln2approx-415.99#.