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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: sophie.tran999 on August 01, 2011, 05:47:57 pm

Title: DIFFERENTIAL EQUATION
Post by: sophie.tran999 on August 01, 2011, 05:47:57 pm: Can anyone help me with this question, i get stuck with a)
Q: in a certain bacterial culture, the rate of increse is proportional to the number of bacteria present.
a) if the number doubles in 3 hours, find the hourly growth rate.
i can deal with b) and c).
Title: Re: DIFFERENTIAL EQUATION
Post by: b^3 on August 01, 2011, 06:21:45 pm: Let N represent the number of bacteria present
let t represent time
then $\frac{dN}{dt}=kN$
so at t=0 lets start of with n Bacteria
then at t=3, N=2n
solve the differential equation
flip it $\frac{dt}{dN}=\frac{1}{kN}$
$t=\int \frac{1}{kN}dN$
$t=\frac{1}{k}log_{e}(|N|)+c$
solve for c, plug in t=0, and N=n $0=\frac{1}{k}log_{e}(|n|)+c\Rightarrow c=-\frac{1}{k}log_{e}(|n|)$
so $t=\frac{1}{k}log_{e}(|\frac{N}{n}|)$
so the hourly growth rate will be k
so plug in t=3 and N=2n
$3=\frac{1}{k}log_{e}(|\frac{2n}{n}|)$
$k=\frac{1}{3}log_{e}(2)$

So the hourly growth rate is $k=\frac{1}{3}log_{e}(2)$
Wait I'm still wrong
Title: Re: DIFFERENTIAL EQUATION
Post by: sophie.tran999 on August 01, 2011, 06:24:11 pm: thx, thats wat i did afetr readin the question 1st time, but look, the question askes to find the GROWTH RATE, not find t in term of N :D
the anwser is : dN/dt = 0.23N
Title: Re: DIFFERENTIAL EQUATION
Post by: b^3 on August 01, 2011, 06:29:14 pm: 1/3loge(2)=0.23 so that means that dN/dt=0.23N, I think that answers it right?
Title: Re: DIFFERENTIAL EQUATION
Post by: b^3 on August 01, 2011, 06:30:18 pm: since growth rate =dN/dt
Title: Re: DIFFERENTIAL EQUATION
Post by: sophie.tran999 on August 01, 2011, 06:32:31 pm: i think so. damn me lol
thx
Title: Re: DIFFERENTIAL EQUATION
Post by: b^3 on August 01, 2011, 06:33:14 pm: Yeh it's right and works, I had myself stuck in a loop for a min there.
Title: Re: DIFFERENTIAL EQUATION
Post by: sophie.tran999 on August 01, 2011, 06:34:09 pm: solve for c, plug in t=0, and N=n, this is important part. i missed this one
instead od plugging t=0 and N=o first, i just pluged t=3 and N=2N. :(