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)
Answer:
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.