# Differential Equation - Radioactive Decay

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• November 4th 2008, 02:06 AM
jeta
Differential Equation - Radioactive Decay
I have a pretty hard differential equation here:

In the nuclear medicine department of a hospital, a radioactive isotope is being produced. The isotope is produced at the rate of 5 milligrams per minute, but has a half-life of only 20 minutes. Write differential equations to find the amount of the isotope present after an hour of production.

Now this one's a bit wierd...

I'm fine with an 'initial amount' problem but when there is a constant production, well it's a bit hard.

What I think to put is

dM/dt = 5-how much is lost (0.5M/20?????)

But I can't really work with that... can someone please help me?
• November 4th 2008, 02:42 AM
mr fantastic
Quote:

Originally Posted by jeta
I have a pretty hard differential equation here:

In the nuclear medicine department of a hospital, a radioactive isotope is being produced. The isotope is produced at the rate of 5 milligrams per minute, but has a half-life of only 20 minutes. Write differential equations to find the amount of the isotope present after an hour of production.

Now this one's a bit wierd...

I'm fine with an 'initial amount' problem but when there is a constant production, well it's a bit hard.

What I think to put is

dM/dt = 5-how much is lost (0.5M/20?????)

But I can't really work with that... can someone please help me?

$\frac{dm}{dt} = 5 - \lambda m$ subject to the boundary condition m = 0 when t = 0 and where $20 = \frac{\ln 2}{\lambda}$.
• November 4th 2008, 02:57 AM
jeta
How do you know that k (you used lamda(sp??)) is equal to ln(2)/Half-life???
• November 4th 2008, 06:20 PM
mr fantastic
Quote:

Originally Posted by jeta
How do you know that k (you used lamda(sp??)) is equal to ln(2)/Half-life???

Read this: Exponential decay - Wikipedia, the free encyclopedia