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I'm currently working on some Differential Calculus, and I'm having a bit of trouble with the following question.

The half-life of an isotope is 150 years. Use this information to determine the differential equation that describes the mass as a function of time. In other words m' = km where k is a constant and m(t) is the mass after t years.

Use the information given to find k, then solve this equation. Use the solution to determine how long it takes for an initial amount to decay to 15% of the original.

For what I know so far, I know little to nothing on the matter. My professor has failed to elaborate on the subject whatsoever, and I've been googling this to try and answer it.

I think I'm supposed to use this question, but I'm not sure if it's the right one...

http://latex.codecogs.com/gif.latex?dm(t)/dt%20=%20-k*m(t)

Any help?

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1 Answer 1

up vote 1 down vote accepted

Hint: If you start with 1 unit of the isotope at time $0$, you have $\frac 12$ after $150$ years. Do you know how to solve $m'=km$? That should give you an equation that (with this data) lets you determine $k$. Now put that $k$ into your solution and find the time when the amount is $0.15$.

Added: your equation is the correct differential equation. The solution is $m(t)=m(0)\exp(-kt)$ as the derivative of $\exp(-kt)$ is $-k \exp(-kt)$

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I don't know how to solve this m′=km exactly. Like I said, my professor did a shoddy job and didn't really explain..anything at all. –  venidi vidici vicini Feb 8 '13 at 3:27
    
Thanks for the help! I used what you said, and came out with 410.5 –  venidi vidici vicini Feb 8 '13 at 3:33
    
@Retrosaur: I didn't fire up the calculator, but a half-life of 150 years says we have one eighth (0.125) left after 450 years, so it looks right. –  Ross Millikan Feb 8 '13 at 3:35
    
Also, thanks for the quick response! –  venidi vidici vicini Feb 8 '13 at 3:37

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