pancha3
2021-01-02
Answered

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.

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Nola Robson

Answered 2021-01-03
Author has **94** answers

Given initial mass:

${m}_{0}=801$ kg at t=0

Let

m(t)=Mass at time t

The exponential function,

$m\left(t\right)={m}_{0}{e}^{kt}$ , where k =constant

Also given:

half life=85days

$\Rightarrow m\left(85\right)=\frac{{m}_{0}}{2}$

Now,

When t=85days:

$m\left(85\right)={m}_{0}{e}^{k\times 85}$

$\Rightarrow \frac{{m}_{0}}{2}={m}_{0}{e}^{85k}$

$\Rightarrow {e}^{85k}=\frac{1}{2}$

$\Rightarrow {e}^{k}={\left(\frac{1}{2}\right)}^{\frac{1}{85}}$

$\Rightarrow k={\mathrm{ln}\left(\frac{1}{2}\right)}^{\frac{1}{85}}$

$\Rightarrow k=-\frac{1}{85}\mathrm{ln}2$

Therefore,

When t=10days:

$m\left(10\right)=801{e}^{k\times 10}$

$=801\left[{e}^{10\times (-\frac{1}{85}\mathrm{ln}2\}}\right]$

$=738.273$ kg

Hence,

The radioactive material remains after 10 days =738.273kg

Let

m(t)=Mass at time t

The exponential function,

Also given:

half life=85days

Now,

When t=85days:

Therefore,

When t=10days:

Hence,

The radioactive material remains after 10 days =738.273kg

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