# Half-Life

Q:  A scientist has 256 g of goo. After 195 minutes, her sample has decayed to 16 g. What’s the half-life of the goo in minutes? (Assume exponential decay)

First we need to set up the problem with our decay formula.  Depending on your school / teacher / book, the variables in the formulas may be different letters, but they all mean the same thing:

P = A*e^(rt)

P is the ending amount, A is the initial amount, r is the growth rate/decay, and t is time.

So, the set-up with our info:

16 = 256*e^(r*195)

Now we need to solve for r:

16 = 256*e^(r*195)

1/16 = e^(r*195)

Take the natural log of both sides:

ln(1/16) = ln(e^(r*195))

Exponents can come down as multipliers (log rules):

ln(1/16) = (195r)*ln(e)

And, ln(e) = 1, so:

ln(1/16) = (195r)

ln(1/16) / 195 = r

Plug into a calculator to get:

r = -.01422

Now, we have r.  We now need to find the half-life.  So, if we started with 256 g, we want to know how long (t) it takes to get to 128 grams.  Plug the info in to our equation:

128 = 256*e^(-.01422*t)

and solve for t:

1/2 = e^(-.01422*t)

ln(1/2) = ln(e^(-.01422*t))

ln(1/2) = -.01422*t

ln(1/2)/-.01422 = t

So, t = 48.74 minutes

The half-life of the goo is approximately 48.74 minutes.