exponents and bacteria

[chiddler]

---

Members don't see this ad.

The question:

After 3 minutes, the number of microbes is reduced from 6.25*10^12 to 2.5*10^11. If this reduction is exponential, then three more minutes would reduce the numbers to how much?

Answer:

Since they were reduced to 1/25th in the first three minuets, then exponential decay would cause another 1/25th to be lost in the next three minutes.

I don't get it. This suggests a linear relationship, doesn't it? If it's exponential, then they must be being reduced at progressively faster rates!

(EK chapter 3, #47)

 

[MT Headed]

This is exponential decay, not exponential growth. It is analogous to half-life in nuclear decay.

The generic formula is X=X0 x b^t
Where X is the current amount
X0 is the initial amount
b is some base (in this case 1/25)
t is some unit of time (in this case 3 minutes)

Every 3 minutes, you take the previous amount and multiply it by 1/25. That's exponential decay. You will never get to 0.

If it was linear decay it would decline by a certain fixed amount every unit of time and eventually reach zero, or even go negative (when it makes physical sense).

 

[chiddler]

This is exponential decay, not exponential growth. It is analogous to half-life in nuclear decay.

The generic formula is X=X0 x b^t
Where X is the current amount
X0 is the initial amount
b is some base (in this case 1/25)
t is some unit of time (in this case 3 minutes)

Every 3 minutes, you take the previous amount and multiply it by 1/25. That's exponential decay. You will never get to 0.

If it was linear decay it would decline by a certain fixed amount every unit of time and eventually reach zero, or even go negative (when it makes physical sense).

Oh I see...

you forgot an "e" in the formula
x=x0*e^bt

thanks for your help

 

[milski]

No, it does not. Linear suggests that for a fixed amount of time their count decreases by a specific amount, for example 50 bacteria/minute. That would get you something like 250, 200, 150, 100... for the first few minutes.

Exponential means that the count for a fixed time changes by a constant proportion, say 1/5. In this case you get 250, 50, 20, 2...

Another way to look at it: if A, B and C are the amount at 0, 1 and 2 minutes, for linear you'll have B-A=C-B while for exponential it will be B/A=C/B. For linear the difference stays the same, for exponential the ratio.

It's also worth noting that the constant in the linear case is a difference with a unit 1/sec while the constant in the exponential case is a ratio and is dimensionless.


---
I am here: http://maps.google.com/maps?ll=47.649721,-122.349802

 

[MedPR]

The question:

After 3 minutes, the number of microbes is reduced from 6.25*10^12 to 2.5*10^11. If this reduction is exponential, then three more minutes would reduce the numbers to how much?

Answer:

Since they were reduced to 1/25th in the first three minuets, then exponential decay would cause another 1/25th to be lost in the next three minutes.

I don't get it. This suggests a linear relationship, doesn't it? If it's exponential, then they must be being reduced at progressively faster rates!

(EK chapter 3, #47)

I did this one last night, and you can answer it a lot easier/faster by taking a look at the data table provided. 🙂