How do I find constants -K and A in a logistics model?

Question

In 2004, there were 1 million facebook users and in 2012, there were 845 million users. Us the model f(t)= 20,000/(20+ae^(-kt)) where t is the number of years since 2004 and f(t) is the number of users in millions to find a and k

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Answer 1 · 2018-04-25T02:53:32

#a=19980#

#k=1.07533329174#

.

#f(t)=20000/(20+ae^(-kt))#

#2012# is #8# years after #2004#, i.e. #t=8#. Therefore,

#f(8)=20000/(20+ae^(-8k))=845# million #color(red)(Equation-1)#

In #2004#, #t=0#

#f(0)=20000/(20+ae^(-(0)k))=1# million #color(red)(Equation-2)#

From #color(red)(Equation-2)#, we have:

#20000=20+ae^0#

#20000=20+a#

#a=20000-20=19980#

From #color(red)(Equation-1)#, we have:

#20000=845(20+ae^(-8k))#

#20000=845(20+19980e^(-8k))#

#20000/845=20+19980e^(-8k)#

#19980e^(-8k)=20000/845-20#

#e^(-8k)=(20000/845-20)/19980#

#e^(-8k)=0.00018361556#

#lne^(-8k)=ln0.00018361556#

#-8klne=ln0.00018361556#

#lne=1#

#-8k=ln0.00018361556#

#k=ln0.00018361556/(-8)=(-8.60266633391)/(-8)#

#k=1.07533329174#