It is known that the mass, m(t), of a radio-active substance decreases as it decays, and that the equation governing this is m'(t)=-0.02 m(t) where t is in years. What is the half-life?

Question

psymon · Answer

$$\frac{ dm }{ dt }= -0.02m$$I think its easier to look at in this way. So you need to do separation of variables. M's must be together on one side by themselves. So you basically just multiply both sides by dt and divide both sides by m to get:
$$\frac{ dm }{ m }=-0.02dt$$ Once you have all the appropriate variables together, you integrate. 
$$\ln(m) = -0.02t+ C$$
$$m = Ce^{-0.02t}$$

Now of course this doesnt tell you the half-life, it just gives you a decay equation. Now, there may be another way to tell, but without trial numbers, the only thing I can say is that the constant of decay, k, always equals:
$$k = \frac{ -\ln2 }{ half-life }$$ Since we were able to get the equation into that decay form of y = ce^(kt), its obvious k is just -0.02. So that means to get thehalf-life we can say:
$$-0.02 = \frac{ \ln2 }{ x } \implies x = \frac{ \ln2 }{ -0.02 } \approx 34.7$$

So that 34.7 would be your half-life. Without being given numbers, I just know that formula. One of the others may be able to tell you more, though.

anonymous · Answer

what is the answer with the approximation? because its not accepting it...

wolf1728 · Answer

Yes, as Psymon said, do you have actual numbers for this?

anonymous · Answer

i don't thats the problem...

psymon · Answer

34.657359

Thatd be the answer with more decimals to it.

anonymous · Answer

The hint I was just given was to find an exponential function which solves the equation. You will the have a formula for the mass, so you can find half-life)

anonymous · Answer

actually psymon that was the correct answer. I guess it only read the exact exact answer instead of the approximation. Thanks!

psymon · Answer

Lol, alright, awesome xD

wolf1728 · Answer

wow - guess it's been a while since I went to school but if the answer is 34.7 is that years, minutes, centuries, etc?

psymon · Answer

Yeah, you can always say k = -ln2/half-life in these growth and decay situtations. So with that knowledge, if you have k or thehalf-life, you can pretty much get everything else. And I suppose this is half-life in years? Who knows, lol.

anonymous · Answer

yea it is in years

wolf1728 · Answer

Yeah I know in advanced math things get more abstract but with this you get a half-life without units?  LOL

psymon · Answer

It gets you half-life under the assumption that t and half-life are in the same units.

wolf1728 · Answer

All right Psymon, that makes a lot more sense.

psymon · Answer

:)