The number of bacteria doubles every 3.5 hours. How many hours does it take for a culture of bacteria to increase from 864 to 442368?

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- Nov 3rd 2008, 04:07 PMmich1992exponential growth
The number of bacteria doubles every 3.5 hours. How many hours does it take for a culture of bacteria to increase from 864 to 442368?

- Nov 3rd 2008, 04:17 PMJameson
This is a question about exponential growth. If N(t) is the number of bacteria after time t, then $\displaystyle N(t)=N_{0}*2^{rt}$, where r is the rate at which the bacteria doubles with respect to t, so in hours. In this problem, it doubles every 3.5 hours, so when t=3.5 we want "rt"=1. That's so we get $\displaystyle N(3.5)=N_{0}*2$ . So to make this expression work, we need rt=1, when t=3.5, so r=(1/3.5). Let's assume that the initial population of bacteria was 1, so our final equation is:

$\displaystyle N(t)=2^{\frac{t}{3.5}}$

Now what you need to do is set this equation equal to 864 and solve for t, do the same for 442368, then calculate the difference in the t's.