Using an exponential function, it is found that it takes 18.6 days for 80% of the material to decay.
The amount of a substance after t days is modeled by:
[tex]A(t) = A(0)e^{-kt}[/tex]
In which:
The half-life is of 8 days, hence [tex]A(8) = 0.5A(0)[/tex], and this is used to find k.
[tex]A(t) = A(0)e^{-kt}[/tex]
[tex]0.5A(0) = A(0)e^{-8k}[/tex]
[tex]e^{-8k} = 0.5[/tex]
[tex]\ln{e^{-8k}} = \ln{0.5}[/tex]
[tex]-8k = \ln{0.5}[/tex]
[tex]k = -\frac{\ln{0.5}}{8}[/tex]
[tex]k = 0.08664339757[/tex]
Hence:
[tex]A(t) = A(0)e^{-0.08664339757t}[/tex]
The time it takes for 80% of the material to decay it t for which [tex]A(t) = 0.2A(0)[/tex], hence:
[tex]0.2A(0) = A(0)e^{-0.08664339757t}[/tex]
[tex]e^{-0.08664339757t} = 0.2[/tex]
[tex]\ln{e^{-0.08664339757t}} = \ln{0.2}[/tex]
[tex]-0.08664339757t = \ln{0.2}[/tex]
[tex]t = -\frac{\ln{0.2}}{0.08664339757}[/tex]
[tex]t = 18.6[/tex]
It takes 18.6 days for 80% of the material to decay.
To learn more about exponential functions, you can take a look at https://brainly.com/question/24906920
