Question

Let x represent number of years. The function P(x)=5x^2+7x+400
represents the population of Town A. In year 0, Town B had a population of 200 people. Town B's population increased by 30 people each year. From year 4 to year 7, which town's population had a greater average rate of change?(1 point)
Responses

Town A
Town A

Town B
Town B
Question 2
Which town will eventually have a greater population?(1 point)
Responses

Town A
Town A

Town B
Town B

Answer 1

1) Town A.
Average rate A = (P(7) − P(4)) / (7−4) = (694 − 508) / 3 = 186 / 3 = 62 people/year.
Town B increases 30 people/year, so A's average rate (62) is greater.

2) Town A.
Compare P_A(x) − P_B(x) = (5x^2+7x+400) − (200+30x) = 5x^2 − 23x + 200.
Discriminant = (−23)^2 − 4·5·200 = 529 − 4000 = −3471 < 0, and leading coefficient 5 > 0, so 5x^2 − 23x + 200 > 0 for all x.
Thus Town A has and will always have the greater population.