Dom V. answered • 09/18/18

Tutor

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Cornell Engineering grad specializing in advanced math subjects

First step is to integrate:

dA/A = k dt

ln(A) = kt + C

_{1}A(t) = C

_{0}e^{kt}At t=0, we have the initial amount of the substance. Call this amount A

_{0}.A(0) = A

_{0}= C_{0}e^{0}= C_{0}, so we know that our constant of integration is simply the initial amount: A(t) = A_{0}e^{kt}.Next we use the second bit of information given in the problem. At t=10 years, we've lost 20% of the starting amount, which leaves 0.8A

_{0}behind:A(10) = 0.8A

_{0}= A_{0}e^{10k}. Now we can solve for k.10k = ln(0.8)

k= (1/10)*ln(0.8).

Finally, we can solve for half life T. At t=T, half the starting amount will remain, A(T)=0.5A

_{0}:0.5 A

_{0}= A_{0}e^{kT}kT = ln(0.5)

T = ln(0.5)/k = 10 ln(0.5)/ln(0.8)