A radioactive material decays according to the function A(t) = 500e^(-.0245t), where A is the amount in grams at the time (t years). What is the half-life of this material? Round your answer to the nearest year.

Question

lgnd_88 · Answer

-0.1386 divided by 2 and rounded

lgnd_88 · Answer

-0.1386/2=-0.0693 rounded to be 700 years. 700 years is the answer

mhchen · Answer

What he said ^^

irishboy123 · Answer

$\large A(t) = 500e^{-\color{blue}{0.0245t}} = 500 \left( \dfrac{1}{2} \right) ^{\dfrac{t}{\tau} } $

where $\large \tau$ is the half-life 

$\large e^{-0.0245 ~ t} =  \left( \dfrac{1}{2} \right) ^{\frac{t}{\tau} } $

$\large -0.0245 ~ t =  \dfrac{t}{\tau}  \ln  \dfrac{1}{2}   $

$\large -0.0245 =  -\dfrac{1}{\tau}   \ln 2 $

$\large \tau =  \dfrac{\ln 2 }{0.0245}   \approx 28.3 $ yrs