Some more practice
There are fast and exact ways to solve these kinds of problems (if you remember your advanced algebra), and there are slower and less exact ways (if you don't). Either way is fine here, although the hints will only be for the non-advanced-algebra route.
What if exactly 3 meningococci surrived the initial onslaught of Frank's immune system? In that case, how long would it be before the critical level of 2.5 million organisms was reached?
(To make this problem interactive, turn on javascript!)
- I need a hint ... :
Remember the equation: N0 * 2^t - ...another hint ... :
In this case, the initial population size (N0) is 3. - ...another hint ... :
If you don't remember advanced algebra, you just have to start guessing. - ...another hint ... :
We know it should be less than 21 generations. - ...another hint ... :
Let's try 15 generations: 3*2^15 = approx 98,000. Not enough. - ...another hint ... :
3*2^19 = about 1.6 million, but 3*2^20 = about 3.1 million.
I think I have the answer: between 19 and 20 generations
The guess-and-try method is fine, if a little slow. However, if you understand how doubling works, you can make some intelligent guesses and speed up your answers. For example, if 2.5 million cells require 21 generations, what would 1.25 million cells require?
With that in mind, try the next problem.
What if Frank was a 10 year old girl with 2.5 liters of blood (as opposed to Frank-the-Flag-Footballer's 5 liters)? In that case, how many generations would it take to get to the lethal level?
(To make this problem interactive, turn on javascript!)
- I need a hint ... :
Remember that the LD50 was based on cells per mL of blood. So, with 1/2 of the blood, the lethal level would be 1/2 of the number of cells. - ...another hint ... :
If we only need 1/2 the cells, that's one less doubling. So instead of 21 generations, we need...
I think I have the answer: 20 generations
One last problem, just to make sure you've got it.
What if Frank had just given blood, losing about half a liter, plus his weakened immune system let 5 meningococci through? Poor Frank!
(To make this problem interactive, turn on javascript!)
- I need a hint ... :
Frank is down to about 4.5 liters of blood - ...another hint ... :
Remember, the LD50 is 500 cells per mL - ...another hint ... :
4.5 liters = 4500 mLs, so the LD50 is about 2.25 million cells - ...another hint ... :
So, 5*2^?? = at least 2.25 million -- try a few numbers...
I think I have the answer: 19 generations gives 2.6 million cells
Copyright University of Maryland, 2007
You may link to this site for educational purposes.
Please do not copy without permission
requests/questions/feedback email: mathbench@umd.edu