Question

The population, P (t), of an Ontario city is modeled by the function p(t) = 14t^2 + 650t + 32,000. If t = 0 corresponds to the year 2,000. When was the population 25,000?

Answer 1

The population of the city was 25,000 in 1970 and 1983.

Explanation:

In order to find the year at which the population was 25,000 we need to make p(t) equal to that number and solve for t as shown below.

`25000 = 14*t^2 + 650*t + 32000\\14*t^2 + 650*t + 7000 = 0\\t^2 + 46.43*t + 500 = 0\\t_(1,2) = (-46.43 \pm √((46.43)^2 - 4*1*500))/(2)\\t_(1,2) = (-46.43 \pm √(155.75))/(2)\\t_(1,2) = (-46.43 \pm 12.48)/(2)\\t_1 = (-33.95)/(2) = -16.98\\t_2 = (-58.91)/(2) =- 29.5`

Since t = 0 corresponds to the year 2000, then t1 = 1983 and t2 = 1970.