A population \( P \) obeys the logistic model. It satisfies the equation \( \frac{d P}{d t}=\frac{5}{900} P(9-P) \) for \( P>0 \). (a) The population is increasing when 0

To determine when the population \( P \) is increasing, we can analyze the expression \(\frac{d P}{d t} = \frac{5}{900} P(9 - P)\). The population increases when \(\frac{d P}{d t} > 0\), which occurs when both factors \( P > 0 \) and \( 9 - P > 0 \) hold true. Therefore, the population is increasing for \( 0 < P < 9 \). This means that as long as the population is positive and under 9, it's experiencing growth. If the population \( P \) were to reach 9, the growth rate would drop to zero, illustrating the carrying capacity of the environment. So remember, keep it under 9 for that growth spurt! In practical terms, understanding this logistic model can help in fields like ecology and resource management. For instance, wildlife officials can use this model to predict population dynamics and establish sustainable harvesting practices. By knowing when a population is thriving (increasing phase), they can determine the best times and methods for intervention, like relocating animals or controlling poaching effectively.