# The half - life of a certain radioactive material is 85 days. An initial amount of the material has a mass of 801 kg Write an exponential function that models the decay of this material. Find how much radioactive material remains after 10 days. Round your answer to the nearest thousandth.

Exponential models
The half - life of a certain radioactive material is 85 days. An initial amount of the material has a mass of 801 kg Write an exponential function that models the decay of this material. Find how much radioactive material remains after 10 days. Round your answer to the nearest thousandth.

2021-01-03
Given initial mass:
$$\displaystyle{m}_{{0}}={801}$$kg at t=0
Let
m(t)=Mass at time t
The exponential function,
$$\displaystyle{m}{\left({t}\right)}={m}_{{0}}{e}^{{{k}{t}}}$$, where k =constant
Also given:
half life=85days
$$\displaystyle\Rightarrow{m}{\left({85}\right)}={\frac{{{m}_{{0}}}}{{{2}}}}$$
Now,
When t=85days:
$$\displaystyle{m}{\left({85}\right)}={m}_{{0}}{e}^{{{k}\times{85}}}$$
$$\displaystyle\Rightarrow{\frac{{{m}_{{0}}}}{{{2}}}}={m}_{{0}}{e}^{{{85}{k}}}$$
$$\displaystyle\Rightarrow{e}^{{{85}{k}}}={\frac{{{1}}}{{{2}}}}$$
$$\displaystyle\Rightarrow{e}^{{k}}={\left({\frac{{{1}}}{{{2}}}}\right)}^{{{\frac{{{1}}}{{{85}}}}}}$$
$$\displaystyle\Rightarrow{k}={{\ln{{\left({\frac{{{1}}}{{{2}}}}\right)}}}^{{{\frac{{{1}}}{{{85}}}}}}}$$
$$\displaystyle\Rightarrow{k}=-{\frac{{{1}}}{{{85}}}}{\ln{{2}}}$$
Therefore,
When t=10days:
$$\displaystyle{m}{\left({10}\right)}={801}{e}^{{{k}\times{10}}}$$
$$\displaystyle={801}{\left[{e}^{{{10}\times{\left(-{\frac{{{1}}}{{{85}}}}{\ln{{2}}}\right\rbrace}}}\right]}$$
$$\displaystyle={738.273}$$kg
Hence,
The radioactive material remains after 10 days =738.273kg