Exponential decay/half-life  TOPIC_SOLVED

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Exponential decay/half-life

Postby KathyInCtrlPA on Mon Mar 04, 2013 7:39 pm

Hi,

I have a general question about whether you can usually use either formula, A(t) = Ae^(kt), or A(t) = Ab^t, in exponential decay (half-life) word problems, or if there is some difference in where each of these formulas is applicable. I'm used to using A(t) = Ae^(kt) for everything, but I saw some resources on the web and videos where A=Ab^t was used (also, A=A(1+r)^t)

For example, in this problem: "Radioactive iodine I 131 has a half life of 8 days . If you start with 20g of radioactive iodine, write a function for the grams of I131 after t days." Can you use either formula for this problem, and if so, would there be a preference?

Please let me know if I can provide any other information, and thank you!

~Kathy
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Re: Exponential decay/half-life

Postby KathyInCtrlPA on Mon Mar 04, 2013 11:03 pm

Ok, I think I understand now that there's an "r" in both the equations... y=a(1+r)^t and y=ae^(rt) and the "r" in each equation represents the growth/decay rate, but the first one is annual (or other fixed period) and the other is continuous, and they're not exactly the same function. Is that right? Also, still not sure how I'm supposed to tell which one to use for the problem I posted.
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Re: Exponential decay/half-life  TOPIC_SOLVED

Postby buddy on Mon Mar 04, 2013 11:58 pm

KathyInCtrlPA wrote:I have a general question about whether you can usually use either formula, A(t) = Ae^(kt), or A(t) = Ab^t, in exponential decay (half-life) word problems, or if there is some difference in where each of these formulas is applicable. I'm used to using A(t) = Ae^(kt) for everything, but I saw some resources on the web and videos where A=Ab^t was used (also, A=A(1+r)^t)

You can convert between them. Forget the "A" for now; just do the e^(kt), b^t, and (1+r)^t.

e^(kt) = (e^k)^t = b^t for b = e^k

(1+r)^t = b^t for b = 1 + r

So you can use any of them. But I've mostly seen e^(kt) for half-life stuff. Unless your book tells you to use b^t, I'd stick with e^(kt). That's also the one they'll want in your science classes. Like if you do chemistry or something. The (1+r)^t is usually really (1 + r/n)^(nt) for n-percent interest compounded n times per year for t years.
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