a) The population is expected to reach a maximum in the year 2004.
b) The maximum population, according to this model, is 72,000 (since the population P(t) is in thousands).
c) population is expected to be 54,000 in the years 1998 and 2010.
How do we find maximum population?
1. P(t)=−0.5(t−14)² + 72,⇒ P(t)=a(t−h)²+k, (h,k) is the vertex of the parabola.
h = 14 k = 72
Year of maximum population = 1990 + h = 1990+14 = 2004
2. Maximum population = k = 72 × 1000 = 72,000
3. population is 54,000, P(t)=54
−0.5(t−14)² + 72 = 54
t₁ = 1998
t₂ = 2010
Year₁ = 1990 + t₁ = 1990 + 1998
Year₂ = 1990 + t₂ = 1990 + 2010