The population of a city is modeled by P(t)=0.5t2 - 9.65t + 100,where P(t) is the population in thousands and t=0 corresponds to the year 2000. a)In what year did the population reach its minimum value? How low was the population at this time?b)When will the population reach 200 000?

Accepted Solution

Answer:Step-by-step explanation:This equation is a positive parabola, opening upwards.  Parabolas of this type have a vertex that is a minimum value.  In order to find the year where the population was the lowest, we have to complete the square to find the vertex.  The rule for completing the square is to first set the parabola equal to 0, then next move the constant over to the other side of the equals sign.  The leading coefficient on the x-squared term HAS to be a positive 1.  Ours is a .5, so will factor it out.  Doing those few steps looks like this:[tex].5(t^2-19.3t)=-100[/tex]Next we take half the linear term, square it, and add it to both sides.  Don't forget the .5 sitting out front there as a multiplier.  Our linear term is 19.3.  Taking half of that gives us 9.65, and 9.65 squared is 93.1225[tex].5(t^2-19.3t+93.1225)=-100+46.56125[/tex]In this process, we have created a perfect square binomial on the left.  Stating that binomial and doing the addition on the right looks like this:[tex].5(t-9.65)^2=-106.8775[/tex]Now finally we will divide both sides by .5 then move over the constant again to get the final vertex form of this quadratic:[tex](t-9.65)^2+106.8775=y[/tex]From this we can see that the vertex is (9.65, 106.8775) which translates to the year 2009 and 107,000 approximately.In our situation, that means that the population was at its lowest, 107,000 in the year 2009.For part b. we will replace the y in the original quadratic with a 200,000 and then factor to find the t values.  Setting the quadratic equal to 0 allows us to factor to find t:[tex]0=.5t^2-9.65t-199900[/tex]If you plug this into the quadratic formula you will get t values of642.02 and -622.72The two things in math that will never EVER be negative are distances/measurements and time, so we can safely disregard the negative value of t.  Since the year 2000 is our t = 0 value, then we will add 642 years to the year 2000 to get thatIn the year 2642, the population in this town will reach 200,000 (as long as it grows according to the model).