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Exponent (Application )

***** CIE 2020 winter Paper 22, No 10****
The number, $b$, of bacteria in a sample is given by $b=P+Q \mathrm{e}^{2 t}$, where $P$ and $Q$ are constants and $t$ is time in weeks. Initially there are 500 bacteria which increase to 600 after 1 week.
(a) Find the value of $P$ and of $Q$.
(b) Find the number of bacteria present after 2 weeks.
[1]
(c) Find the first week in which the number of bacteria is greater than 1000000 .
[3]
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*********math solution*************
(a) When $t=0, P+Q=500 \cdots (1)$ When $t=1, P+Qe^2=600\cdots (2)$ $(2)-(1): Qe^2-Q=100\Rightarrow Q(e^2-1)=100.$ Hence $Q=\dfrac{100}{e^2-1}=15.65$ and $P=500-15.65=484.35$
Ans $P=484,Q=15.7$
(b) When $t=2, B=484.35+15.65e^4=1338.$
(c) Since $B>1000000, 484.35+15.65e^{2t}>1000000$. Thus $e^{2t}>\dfrac{1000000-484.35}{15.65}$ Therefore $2 t>\ln \left(\dfrac{1000000-484.35}{15.65}\right) \Rightarrow t>5.5$
Ans: 6th week.
**********end math solution********************

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