the population of a certain region grew from 18.1 million in 2005 to 21.9 million in 2012. a. find a…
the population of a certain region grew from 18.1 million in 2005 to 21.9 million in 2012. a. find a function of the form ( p(t)=ce^{kt} ) that models the population growth of the region. here, ( t ) is the number of years since 2005, and ( p(t) ) is in millions. round your ( k ) value to 3 decimal places. ( p(t)= ) b. use your model to predict the population of this region (in millions of people) in 2019. round your answer to the nearest tenth of a million.
Answer
Explanation:
Step1: Find the value of (C)
When (t = 0) (year 2005), (P(0)=18.1). Substitute into (P(t)=Ce^{kt}), we get (P(0)=Ce^{k\times0}=C). So (C = 18.1).
Step2: Find the value of (k)
When (t = 7) (year 2012), (P(7)=21.9). Since (C = 18.1), the equation becomes (21.9=18.1e^{7k}). First, divide both sides by (18.1): (\frac{21.9}{18.1}=e^{7k}). Then take the natural - logarithm of both sides: (\ln(\frac{21.9}{18.1})=\ln(e^{7k})). Using the property (\ln(e^{x})=x), we have (\ln(\frac{21.9}{18.1}) = 7k). Calculate (\ln(\frac{21.9}{18.1})\approx\ln(1.209945)\approx0.191). So (k=\frac{\ln(\frac{21.9}{18.1})}{7}\approx\frac{0.191}{7}\approx0.027).
The population model is (P(t)=18.1e^{0.027t}).
Step3: Predict the population in 2019
For the year 2019, (t = 14) (since (2019 - 2005=14)). Substitute (t = 14) into (P(t)=18.1e^{0.027t}), we get (P(14)=18.1e^{0.027\times14}). First, calculate (0.027\times14 = 0.378). Then (e^{0.378}\approx1.459). So (P(14)=18.1\times1.459\approx26.4).
Answer:
a. (P(t)=18.1e^{0.027t}) b. (26.4)