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Say a culture of bacteria doubles in weight every 24 hours. If it originally weighed 10g, what would be its weight after 18 hours?

I know how to calculate half-life but don't know about doubling time. What is the easiest formula to use in order to solve this?

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Since the initial weight is $10$g, and the weight doubles every $24$ hours, you know that after $t$ hours the weight will have doubled $\frac{t}{24}$ times. Thus, if $w(t)$ is the weight after $t$ hours, we must have

$$w(t)=10\cdot2^{t/24}\;.\tag{1}$$

You want $w(18)$, so just substitute $t=18$ into $(1)$.

Note that this is exactly the way that half-lives behave, but with a power of $2$ (for doubling) instead of a power of $\frac12$ (for halving). That is, if the half-life were $24$ hours, the weight after $t$ hours would be

$$10\left(\frac12\right)^{t/24}\;.$$

Thus, it’s not really anything new. You should be able to work just as will with a tripling time, or the time it takes for a reduction to one-tenth of the weight, or any similar way of giving a rate of geometric growth or decay.

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  • $\begingroup$ For calculating half-lives, I like to use the formula ln(1/2)= -kt1/2. So you're saying that I can just replace that 1/2 with a 2 for doubling time? $\endgroup$
    – jaykirby
    Jul 20, 2013 at 3:46
  • $\begingroup$ @jaykirby: I’m not sure how to read your -kt1/2. In any case, the present problem isn’t analogous to finding a half-life, since you’re given the doubling time. $\endgroup$
    – Brian M. Scott
    Jul 20, 2013 at 3:49
  • $\begingroup$ so what formula would you recommend I use for calculating doubling time? $\endgroup$
    – jaykirby
    Jul 20, 2013 at 3:52
  • $\begingroup$ @jaykirby: Depends on what data you’re given. Exactly what kind of problem do you have in mind? $\endgroup$
    – Brian M. Scott
    Jul 20, 2013 at 3:58
  • $\begingroup$ Same kind of problem like the one here. $\endgroup$
    – jaykirby
    Jul 20, 2013 at 4:01

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