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# The mass of a radioactive isotope at time t (in years) is $M(t)=200 e^{-0.14 t} \mathrm{g}$. What is the mass of the isotope initially? How fast is the mass of the isotope changing 2 years later?

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To find the initial mass of the isotope we will compute $M(0)$.

$\pmb{M(0)}=200\cdot e^{-0.14\cdot 0}=\pmb{200}$

We conclude that the initial mass of the isotope is $200\mathrm{\,\,g}$.

To find how fast the mass of the isotope is changing after $2$ years we will compute $M^\prime(2)$.

\begin{align*} M^\prime(t)&=(200\cdot e^{-0.14t})^\prime\\ &=200\cdot e^{-0.14t}\cdot (-0.14t)^\prime\\ &=-200\cdot 0.14\cdot e^{-0.14t}\\ &=28\cdot e^{-0.14t} \end{align*}

Substituting $t=2$ in $M^\prime(t)$ we get

$\boxed{M^\prime(2)=-21.16}$

We conclude that the mass of the isotope is changing approximately $-21\mathrm{\,\, g}$ per year.

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