Suppose the population of a town was 40,000 on January 1, 2010 and was 50,000 on January 1, 2015. Let P(t) be the population of the town in thousands of people t years after January 1, 2010.6 (a) Build an exponential model (in the form P(t) = a bt ) that relates P(t) and t. Round the value of b to 5 significant figures.a = ?b = ?

Answer:Given,The initial population ( on 2010 ) = 40,000,Let r be the rate of increasing population per year,Thus, the function that shows the population after t years,[tex]P(x)=40000(1+r)^t[/tex]And, the population after 5 years ( on 2015 ) is,[tex]P(5)=40000(1+r)^{5}[/tex]According to the question,P(5) = 50,000,[tex]\implies 40000(1+r)^5=50000[/tex][tex](1+r)^5=\frac{50000}{40000}=1.25[/tex][tex]r + 1= 1.04563955259[/tex][tex]\implies r = 0.04653955259\approx 0.04654[/tex]So, the population is increasing the with rate of 0.04654,And, the population after t years would be,[tex]P(t)=40000(1+0.04654)^t[/tex][tex]\implies 40000(1.04654)^t[/tex]Since, the exponential function is,[tex]f(x) = ab^x[/tex]Hence, by comparing,a = 40000,b = 1.04654