Question

Portland's population in 2007 was about 568 thousand, and had been growing by

Answer 1

Answer:

$a) \ \ P_n=(1+r)P_{n-1}\\\\b)\ \ \ P_n=1.011^n(568000)\\\\c)\ \ 626,771\\\\d)\ \ 19.10yrs\ later \ (2027 \ February)$

Step-by-step explanation:

a. Given that the population starts at 568000 and grows at 1.1% each years.

-The recursive formula for the population takes the form:

$P_n=(1+r)P_{n-1}$

Where:

$P_n$ is the population at the nth year.

$r$ is the rate of growth

$P_{n-1}$ is the population a year before the nth year.

Hence, the recursive formula is given by $P_n=(1+r)P_{n-1}$

b. The explicit formula of a population growth takes the form:

$P_n=(1+r)^nP_o$

-Given that r=1.1% and the initial population is 568000

$P_n=(1+r)^nP_o\\\\r=1.1\%=0.011\\\\P_n=(1+0.011)^nP_o, P_o=568000\\\\P_n=1.011^n(568000)$

Hence, the explicit formula is $P_n=1.011^n(568000)$

c. The population in 2016 can be determined using the explicit formula.

-We substitute the growth rate and initial population as follows:

$P_n=(1+r)^nP_o\\\\=1.011^n(568000)\\\\n=2016-2007=9\\\\\therefore P_{2016}=(1+0.011)^9\times 568000\\\\=626,770.7721\approx626,771$

Hence, the population in 2016 will be approximately 626,771

d. Given that the population after n years will be 700000.

#We substitute this value in the explicit formula to solve for n then add it to the initial year, 2007;

$P_n=(1+r)^nP_o\\\\700000=1.011^n(568000)\\\\1.011^n=\frac{700000}{568000}=\frac{700}{568}\\\\n=\frac{log \ (700/568)}{log\ 1.011}\\\\=19.10\ years\\\\\#Add \ 19.10yrs \ to \ 2007\\\\=2007+19.10yrs\\\\=2026.10\approx2027$

Hence, the population will get to 700000 after 19.10 years or in February the year 2027