# The half-life of a certain radioactive material is 85 days. An initial amount of the material has a mass of 801 kg. How do you write an exponential function that models the decay of this material and how much radioactive material remains after 10 days?

Jul 18, 2016

Let
${m}_{0} = \text{Initial mass"=801kg " at } t = 0$

$m \left(t\right) = \text{Mass at time t}$

$\text{The exponential function} , m \left(t\right) = {m}_{0} \cdot {e}^{k t} \ldots \left(1\right)$
$\text{where " k=" constant}$

$\text{Half life} = 85 \mathrm{da} y s \implies m \left(85\right) = {m}_{0} / 2$

Now when t =85days then

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

$\implies {m}_{0} / 2 = {m}_{0} \cdot {e}^{85 k}$

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

Putting the value of ${m}_{0} \mathmr{and} {e}^{k}$ in (1) we get

$m \left(t\right) = 801 \cdot {2}^{- \frac{t}{85}}$ This is the function.which can also be written in exponential form as

$m \left(t\right) = 801 \cdot {e}^{- \frac{t \log 2}{85}}$

Now the amount of radioactive material remains after 10 days will be

$m \left(10\right) = 801 \cdot {2}^{- \frac{10}{85}} k g = 738.3 k g$