In a research project, $5\,000$ ants are put in a man-made habitat for the population to grow. Aft

Question

In a research project, $5\,000$ ants are put in a man-made habitat for the population to grow. After $t$ days, the population of the ants, $P$ (in thousands) is given by $$P = -45e^{At} + B, \text{ for some constants}\hspace{0.5em} A \text{ and}\hspace{0.5em} B.$$ Given that $10$ days later, the population has increased to $35\,000$. Show that $B = 50$ and $A = k \ln 3$, where $k$ is a constant to be found.[4] Find the population of the ants after $20$ days.[2] Find the population of the ants after a very long time.[1] Sketch the graph of $P$ against $t$.[1]

Accepted Answer

(i) $B=50$, $A=-\dfrac1{10}\ln3$, $k=-\dfrac1{10}$ (ii) $45\,000$ ants (iii) Approaches $50\,000$ ants (iv) Increasing curve from $(0,5)$, with horizontal asymptote $P=50$.

2013 JJC H1 Promo Q4

2013 JJC H1 Promo Q4

Solution:

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