Half-Life College Algebra

Question

A substance has a half-life of  minutes. If the initial amount of the substance was  grams, how many half-lives will have passed before the substance decays to  grams. What is the total time of the decay?

Answer 1

We can use exponential decay to model the half life.

Recall that N(t) = N₀e^-dt.

A quick glossary:

1. t is the time in minutes.
2. N(t) is amount of the substance after t time has passed.
3. N₀ is the original amount of the substance.
4. e is the exponential base, it has inverse ln.
5. d is the rate of decay.

Since our first half life is 2.671 minutes. We know that (0.5)N₀ = N(2.671) = N₀e^-d(2.671).

So we compute:

(0.5)N₀ = N₀e^-d(2.671)

(0.5)= e^-d(2.671)

ln(0.5)= -d(2.671)

-ln(0.5)= d(2.671)

-ln(0.5)/2.671= d

We now have our rate of decay (d) which is approximately 0.2595. We will revisit the exponential decay model, N(t) = N₀e^-dt and solve for t when N(t) = 8.9.

We compute:

8.9 = N(t) = 142.4e^-0.2595t

8.9/142.4 = e^-0.2595t

ln(8.9/142.4) = -0.2595t

-ln(8.9/142.4) / 0.2595= t

10.6843 = t

We can estimate to see if our solution makes sense. Since our half life is about 2.5 minutes, and we needed about 10 minutes, there are 4 half life intervals. Which means that we are going to halve our original amount 4 times AKA we will divide by 2⁴ = 16. Since 142/ 16 = 8.875, our answer should be correct.