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Answer :
It will take approximately 350.52 years for only 215 kg of the sample to remain.
The function that models the amount of the isotope after t years is given by:
A(t) = 1235 × [tex]0.5^{(t/50)}[/tex]
where A(t) is the amount of the isotope after t years and 1235 is the original amount.
To find the amount of the isotope after 421 years, we can plug in 421 for t in the function above:
A(421) = 1235 × [tex]0.5^{(421/50)}[/tex] = 12.41
Thus, after 421 years, only 12.41 kg of the isotope will remain.
To find the time at which only 215 kg of the sample will remain, we can solve the equation:
215 = 1235 × [tex]0.5^{(t/50)}[/tex]
for t. This gives us:
t = 50 × log(215/1235) ÷ log(0.5) = 350.52
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