The population, $P$ thousand, of a certain species of bats in a cave is modelled by \[P = a - b\ma

Question

The population, $P$ thousand, of a certain species of bats in a cave is modelled by $$P = a - b\mathrm{e}^{-ct},$$ where $t$ is the time in years and $a$, $b$ and $c$ are constants. When $t = 0$ the population is $7000$ and the rate of increase of population is $540$ per year. The population approaches $25\,000$ in the long run. Find the values of $a$, $b$ and $c$.[4] Sketch the graph of $P$ against $t$.[1] The population, $Q$ thousand, of another species of bats in another cave is modelled by $$Q = 3\sqrt{t} + \frac{150}{\sqrt{t^3}}, t > 0.$$ Using differentiation, find the $t$-coordinate of the stationary point on the graph of $Q$ against $t$ and determine the nature of the stationary point.[4] Find the duration of time, in years, for which the population $P$ is more than the population $Q$.[2]

Accepted Answer

(a)a=25, b=18, c=0.03 (c)$t=5\sqrt 6$, minimum point (d) 20.5 years

2024 VJC H1 Q6

2024 VJC H1 Q6