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?

Step-by-step explanation:

Use the half life formula

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:

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

Now take the natural log of both sides since the natural log and that e are inverses. The e then disappears:

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

Divide both sides by 70 to get the k value of

k = -.00769995

Now the second equation looks like this:

where t is our only unknown now that we know k.

Begin by dividing both sides by 12 to get

and take the natural log of both sides again to eliminate the e:

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

answer: the answer is 35%