The rate of decay in the mass, M, of a radioactive substance is given by the differential equation dM /dt = -kM, where k is a positive constant. If the initial mass was 200g, then find the expression for the mass, M, at any time t.

- anonymous

- katieb

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- anonymous

- anonymous

\[\frac{ 1 }{ -k }\]\[\int\limits \frac{ dM }{ M } = \int\limits dt\]

- anonymous

is that right?

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## More answers

- anonymous

Yea you mean \[\frac{ -1 }{ k} \times \int\limits \frac{ dM }{ M}=\int\limits dt\] R
Right?

- anonymous

integrate both side and it is \[\frac{ -1 }{ k}( \ln M)=t\]

- anonymous

okay how did you integrate the left side

- anonymous

okay nvm i see what you did

- anonymous

Cross multiply so you can get M by it self.
\[-1(\ln M)=kt\]

- anonymous

now i just solve for M right?

- anonymous

ln M=-kt

- anonymous

M = e ^-kt

- anonymous

e both sides and it should look
\[M=e ^{-kt}\]

- Astrophysics

Don't forget the constant you will need it for your initial amount

- anonymous

Yea integrate both side and solve for M because he want to find an expression for M

- anonymous

200 = e^-k(0) ...?

- anonymous

Sorry had trouble with equation table.

- anonymous

these are my answer choices by the way
M = 200ln(kt)
M = 2e−kt
M = 200 ekt
M = 200 e−kt

- anonymous

is it D

- Astrophysics

yes

- anonymous

Yes, because if it is M=200e^-kt

- anonymous

okay thanks for the help guys :D

- anonymous

if it is M=200e^-kt*

- anonymous

you multiply it by 200 since it is decaying and it is the initial principal.

- Astrophysics

\[\frac{ dM }{ dt } = -kM \implies \int\limits \frac{ d M}{ M } = \int\limits - k dt \]
\[\ln(M) = -k t+C\]
\[\large e^{\ln(M)} = e^{-kt+C}\]
\[M = e^{-kt}e^C\]
let \[e^c = C\]
so we have \[M = Ce^{-kt}\] after putting our initial conditions we get \[M = 200e^{-kt}\]

- Astrophysics

Just to note initial conditions is t = 0 meaning M = 200, so C = 200

- anonymous

This method works too, but I just multiply the initial mass amount by the expression.

- anonymous

Since he probably is taking a calculus class with some basic differential equation problems. From my experience, it is similar to when I took Calculus. Doubt this is a Differential Equation class questions. I never took DE.

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