The population increase of a city

Question

the populations P (in thousands) of a certain city from 2000 through 2007 can be modeled by P=1666.9e^kt, Where t represent the year, with t= 0 corresponding to 2000. In 2006 the population of the city was about 1,930,000.

a). find the value of K (round your answer to five decimal places)

k=

b). is the population increasing or decreasing?

c). us the model to find the population of the city in 2010 to 2015 (round the the nearest three decimal places)

2010 p=_________ thousand

2015 p=_________ thousand

d). according to the model what year will the population reach 2.5 million?

Answer 1

1930 = 1666e^6k divide both sides by 1666

1930/1666 = e^6k take natural logs of both sides

ln(1930/1666) = 6k divide by 6 and punch the numbers into a calculator with a log function

k= ln(1930/1666)/6 = 0.0245

k = 0.0245 = about a 2.45% annual growth rate

k>0 so the population is increasing

for 2010 t = 10 P = 1666e^.245 = 2128.517 thousands = 2,128,517

for 2015 t = 15 P = 1666e^(.0245)(15) = 1666e^.3675 = 2405.904 thousands = 2,405,904

it reaches 2.5 million in probably about a year later than 2015 which had about 2.4 million

2,500 = 1666e^.0245t

2500/1666 = e^.0245t

ln(2500/1666) = .0245t

t = ln(2500/1666)/.0245 = about 16.6 years or the year 2016 plus half a year = 2017