You order two bacterial strains from Amazon: L. acidophilous to make yogurt, and R. japonicum to help your vegetable garden. According to the website, L acidophilous has a doubling time of 66 minutes, while R. japonicum has a doubling time of 344 minutes. But when you receive the package, you find that only one strain has been included, and its label was mangled so that it only reads "N_t = N₀ * e^{(0.002 t)}". Should you put this one in the yogurt, or dig it into your garden?

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• I need a hint ... : what is the growth rate, m?
  
  
• ...another hint ... : you need to solve the equation m = ln(2)/doubling time to find out what the doubling time is.
  
  

After putting the Rhizobium in your garden, you don't notice much difference for several days. After a few irate phone calls, you discover that you applied the Rhizobium at 1/100 of the necessary strength. Given the growth rate of Rhizobium above, how long will it take for Rhizobium to get up to a useful level

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• Try an estimate first: how many doubling times? between 6 and 7. So, 6.5 * 344 minutes = 2236 minutes, or 37 hours
  
  
• Now try an exact answer : Increasing 100 fold is the same as setting N₀ to 1 and N_t to 100.
  
  
• Plug and chug... : 100 = 1 * e^(0.002t)
  
  
• take the ln of both sides : ln 100 = 0.002 t
  

But wait, the conditions in your garden are hardly optimal. In fact, the doubling time for R. japonicum is about 10 times longer than in the lab. How does this affect the time you need in your garden? And what is the new growth rate m?

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• I need a hint ... : You still need the same number of doublings to get to a 100-fold increase
  
  
• ...another hint ... : But each doubling takes 10 times as long
  
  
• ...another hint ... : So you need 10 times as much time, that is, 384 hours, or 16 days.
  
  
• ...the new growth rate... : Use m = ln(2)/doubling time
  
  
• ...another hint ... : if doubling time got 10 times bigger, m must have gotten 10 times smaller
  
  

So after 8 hours in the garden vs. 8 hours on the labbench, starting with 2.5 million Rhizo each, how many Rhizo do you expect?

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• on the lab bench... : N_t = 2.5 million * e ^(0.002*8*60)
  
  
• in the garden... : N_t = 2.5 million * e ^{(0.0002*8*60)}
  
  

You decide to measure the growth rate in your garden. You culture the Rhizo at three different times, with the following results:

time	number cells (x106)
day 0	1.2
day 7	4.0
day 14	13.5

 

What is the actual rate of growth (m) in your garden?

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• I need a hint ... : Find the doubling time first
• ...another hint...: m = ln(2)/doubling time