A substance with a half life is decaying exponentially. If there are initially 12grams of the substance and after 70 minutes there are 7 grams, after how many minutes will there be 2 grams remaining?

[tex]N=N_0e^{kt}[/tex] where N is the amount after decay, No is the initial amount, e is Euler's number, k is the decay constant, and t is the time in minutes. For us, the first equation looks like this:

[tex]7=12e^{k(70)}[/tex] and we will solve that for k and then use that value of k in the second equation to find time.

Begin by dividing 7 by 12 to get

[tex]\frac{7}{12}=e^{k(70)}[/tex] Now take the natural log of both sides since the natural log and that e are inverses. The e then disappears:

[tex]ln(\frac{7}{12})=k(70)[/tex]

Plug the left side into your calculator and then set it equal to the right side, giving us:

[tex]-.5389965007=70k[/tex]

Divide both sides by 70 to get the k value of

k = -.00769995

Now the second equation looks like this:

[tex]2=12e^{(-.00769995)t[/tex] where t is our only unknown now that we know k.

Begin by dividing both sides by 12 to get

[tex]\frac{1}{6}=e^{-.00769995t[/tex] and take the natural log of both sides again to eliminate the e:

[tex]ln(\frac{1}{6})=-.00769995t[/tex]

Take care of the left side on your calculator to get

-1.791759469 = -.00769995t and divide both sides by -.00769995 to get

t = 232.697 minutes