Question

The rate of decay in the mass, M, of a radioactive substance is given by the differential equation dM dt equals negative 1 times k times M, where k is a positive constant. If the initial mass was 150g, then find the expression for the mass, M, at any time t.

Answer 1

https://gyazo.com/8d62d90df17bfc829809548f15163bd1

Answer 2

@Zarkon

Answer 3

Can you confirm my answer?

Answer 4

@baru Can you confirm for me please?

Answer 5

@jim_thompson5910 will you be my hero?

Answer 6

If \(\Large M = 150e^{-kt}\) then what is \(\Large \frac{dM}{dt}\) equal to?

Answer 7

-kM

Answer 8

@jim_thompson5910 when I integrated I got M=e^-kt

Answer 9

I wasn't sure what to do with the initial mass, so using the choices I multiplied it.

Answer 10

Would you like to see my work?

Answer 11

sure, go ahead and post your work

Answer 12

Please ignore the 100-60 crap, that was for another question.

Answer 13

You should have M in absolute values. And don't forget the +C

\[\Large \ln(|M|) = -kt+C\]

\[\Large |M| = e^{-kt+C}\]

\[\Large |M| = e^{C}*e^{-kt}\]

Answer 14

Why in absolute values?

Answer 15

because the domain of ln(x) is x > 0

the absolute values ensure that the argument is not negative

Answer 16

Ahh ok.

Answer 17

Would e^c be equal to 150?

Answer 18

now you'll use the information that 150 is the initial mass

so M = 150 when t = 0


\[\Large |M| = e^{C}*e^{-kt}\]

\[\Large |150| = e^{C}*e^{-k*0}\]

\[\Large |150| = e^{C}*e^{0}\]

\[\Large |150| = e^{C}*1\]

\[\Large |150| = e^{C}\]

\[\Large 150 = e^{C}\]

\[\Large e^{C} = 150\]

you are correct

Answer 19

Thank you good sir.

Answer 20

Now we make the claim that \[\Large M = 150e^{-kt}\]

here is how to check that claim


If \[\Large M = 150e^{-kt}\] then \[\Large \frac{dM}{dt} = -150ke^{-kt}\]


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Start with the initial differential equation and perform substitutions


\[\Large \frac{dM}{dt} = -k*M\]


\[\Large -150ke^{-kt} = -k*150e^{-kt}\]


\[\Large -150ke^{-kt} = -150ke^{-kt} \ \ \ \color{green}{\checkmark}\]


So the solution has been confirmed

Answer 21

Of course, the more general solution is \[\Large M = Ce^{-kt}\] but we don't need to worry about that

Answer 22

Got it. Thanks