Question

A radioactive element decays according to the function y=y0 e −0.0307t​, where t is the time in years. If an initial sample contains y0 = 8 grams of the​ element, how many grams will be present after 20 ​years? What is the​ half-life of this​ element?

Answer 1

Given that an initial sample contains y0 = 8 grams of the​ element,

Then grams that will be present after 20 ​years can be calculated by using given decay function

y=y0 e^-0.0307t

We have y0= 8 grams

And t = 20 years

Then y= 8 e^{-0.0307× 20}

on taking logarithm both sides

ln y= ln(8) -0.0307×20

ln y= 1.4654

Take antilogarithm

y= antilog(1.4654)

y= 4.329 grams

Next , in half life , amount of element left half

Then y0/2 = y0 e-0.0307t

On taking logarithm we get

ln2= 0.0307 t

t= ln2/0.0307

t= 22.578 years

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