he populations P (in thousands) of a certain city from 2000 through 2008 can be modeled by P = 287.82ekt, where t is the year, with t = 0 corresponding to 200

Question

he populations P (in thousands) of a certain city from 2000 through 2008 can be modeled by
P = 287.82ekt,
where t is the year, with t = 0 corresponding to 2000. In 2006, the population was about 368,000.

 

(a) Find the value of k for the model. Round your result to four decimal places.
k =




(b) Use your model to predict the population in 2015. (Round your answer to the nearest person.)
P =

Answer 1

Part a:

Since 2006 is 6 years after 2000, t = 6.  The population of 368000 means P = 368 since population is measured in thousands.

 

368=287.82e^k(6)

368=287.82e^6k

1.2786=e^6k

ln(1.2786)=ln e^6k

ln(1.2786)=6k

ln(1.2786)/6=k

k=0.0410

 

Part b:

2015 is 15 years after 2000 so t=15.  We use k=0.0410 from the previous problem.

 

P=287.82e0.0410(15)=287.82e^0.615=532.461 thousands = 532461 people.

Answer 2

368,000 = 287.82e^6k

368,000/287.82 = e^6k

ln(368,000/287.82) = 6k

k = (ln(368,000/287.82))/6

 

Whatever value you get, plug that, and t = 15 in.