Question

Pleasantburg has a population growth model of P(t)=at2+bt+P0 where P0 is the initial population. Suppose that the future population of Pleasantburg t years after January 1, 2012, is described by the quadratic model P(t)=0.9t2+6t+14,000. In what month and year will the population reach 21,100?

Answer 1

Answer: In June 2097

Step-by-step explanation:

According to the model, to find how many years t should take for  $P(t)=21100$ we must solve the equation  $0.9t^2+6t+14000=21100$. Substracting 21100 from both sides, this equation is equivalent to $0.9t^2+6t-7100=0$.

Using the quadratic formula, the solutions are $t_1= \frac{-6-\sqrt{6^2 -4*0.9*(-7100)}}{2*0.9}=-92.21$ and $t_2=\frac{-6+\sqrt{6^2 -4*0.9*(-7100)}}{2*0.9}=85.54$. The solution $t_1=-92.21$ can be neglected as the time t is a nonnegative number, therefore $t=t_2=85.54$.

The value of t is approximately 85 and a half years and the initial time of this model is the January 1, 2012. Adding 85 years to the initial time gives the date  January 2097, and finally adding the remaining half year (six months) we conclude that the date is June 2097.