Respuesta :
Using an exponential function, it is found that it will take 3,252 years for the sample to degrade to 4.5kg.
What is the exponential function for the amount of a substance?
It is given by:
[tex]A(t) = Pe^{-kt}[/tex]
In which:
- P is the initial amount.
- k is the decay rate, as a decimal.
The half-life is of 21,400 years, hence, A(21400) = 0.5P, and this is used to find k.
[tex]A(t) = Pe^{-kt}[/tex]
[tex]0.5P = Pe^{-21400k}[/tex]
[tex]e^{-21400k} = 0.5[/tex]
[tex]\ln{e^{-21400k}} = \ln{0.5}[/tex]
[tex]-21400k = \ln{0.5}[/tex]
[tex]k = -\frac{\ln{0.5}}{21400}}[/tex]
k = -0.00003239005
Hence, considering the initial mass is of P = 5 kg, the amount after t years is given by:
[tex]A(t) = 5e^{-0.00003239005t}[/tex]
The amount will be of 4.5 kg at t for which A(t) = 4.5, hence:
[tex]4.5 = 5e^{-0.00003239005t}[/tex]
[tex]e^{-0.00003239005t} = 0.9[/tex]
[tex]\ln{e^{-0.00003239005t}} = \ln{0.9}[/tex]
[tex]-0.00003239005t = \ln{0.9}[/tex]
[tex]t = -\frac{\ln{0.9}}{0.00003239005}[/tex]
t = 3252.
More can be learned about exponential functions at https://brainly.com/question/25537936
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