Question

9. Determine how long it takes for 80% of the material to decay.

Answer 1

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:

$A(t) = A(0)e^{-kt}$

In which:

• k is the decay rate, as a decimal.

• A(0) is the initial amount.

The half-life is of 8 days, hence $A(8) = 0.5A(0)$, and this is used to find k.

$A(t) = A(0)e^{-kt}$

$0.5A(0) = A(0)e^{-8k}$

$e^{-8k} = 0.5$

$\ln{e^{-8k}} = \ln{0.5}$

$-8k = \ln{0.5}$

$k = -\frac{\ln{0.5}}{8}$

$k = 0.08664339757$

Hence:

$A(t) = A(0)e^{-0.08664339757t}$

The time it takes for 80% of the material to decay it t for which $A(t) = 0.2A(0)$, hence:

$0.2A(0) = A(0)e^{-0.08664339757t}$

$e^{-0.08664339757t} = 0.2$

$\ln{e^{-0.08664339757t}} = \ln{0.2}$

$-0.08664339757t = \ln{0.2}$

$t = -\frac{\ln{0.2}}{0.08664339757}$

$t = 18.6$

It takes 18.6 days for 80% of the material to decay.

To learn more about exponential functions, you can take a look at brainly.com/question/24906920