Question

The populations P (in thousands) of a certain city from 2000 through 2008 can be modeled by P = 1118.8ekt, where t is the year, with t = 0 corresponding to 2000. In 2002, the population was about 1,400,000. (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 = thousand

Answer 1

I guess that the function is \[P(t)=1118.8e^{kt}\]if it is, we have to replace 2 on t and find the value of k\[P(2)=1118.8e^{2k}\]\[P(2)=1,400,000=1118.8e^{2k}\]\[1,400,000=1118.8e^{2k}\]divide both sides by 1118.8\[1251.34=e^{2k}\]then apply natural logarithms at both sides\[ln(1251.34)=ln(e^{2k})\]\[7.1320=2k\\\boxed{\boxed{k=3.566}}\]now we know the function, and we just have to replace t=15\[P(t)=1118.8e^{3.566t}\]\[P(15)=1118.8e^{3.566\cdot15}\]\[P(15)=39,577.338\]now we have to multiple by 1000\[\boxed{P(15)=39,577,338}\]