Expert: Paul Klarreich - 10/29/2006

Question
At the beginning of an experiment, a scientist has 264 grams of radioactive goo. After 240 minutes, her sample has decayed to 33 grams.

Find a formula for G(t), the amount of goo remaining at time t. G(t) =?

Answer
Questioner:  Gur
Category:  Calculus
 
Subject:  radioactive decay problem
Question:  At the beginning of an experiment, a scientist has 264 grams of radioactive goo.

>> That's spelled  Gu, with a capital G.

After 240 minutes, her sample has decayed to 33 grams.

Find a formula for G(t), the amount of goo remaining at time t. G(t) =?
.....................................................
Hi, Gur,

The general rule here is that the amount G(t) is given by the general rule:

G(t) = G0 e^(-Lt)

which has two parameters:

G0 = the amount at the start, i.e. at  t=0.
L  = some coefficient that is related to the half-life of the isotope.  [Gu-211, I assume]

So your task is to determine G0 and L from the given conditions.  Since you have two variables to find, you need two facts, such as:

At  t = 0,  you have  264 grams.  G(0) = 240
At  t = 4 hours (240 minutes) you have  33 grams.   G(4) = 33

So just substitute:

G(0) = 240 = G0 e^(-L(0))

240 = G0 e^0
240 = G0    << Got the first one.

G(4) = 33 = 240 e^(-L(4))

33 = 240 e^(-4L)

e^(-4L) = 33/240 = 11/80

Now logarithim-ize that:

-4L =  ln(11/80)

L = - ln(11/80)/4

and fire up your calculator for that.  Mine gives:

L = 0.49603284046887776705193606027344, or about:  L = 0.496

So your final equation is:

G(t) = 240 e^(-0.496t)