Precalculus one

Question

At the beginning of an experiment, a scientist has 312 grams of radioactive goo. After 60 minutes, her sample has decayed to 9.75 grams. 

What is the half-life of the goo in minutes?

Answer 1

Nothing wrong with Mark's answer.

9.75 and 312 were chosen for a reason, though.

9.75/312 = 1/32 = 1/2⁵

Then, as there are 60 minutes in an hour, and 5 half-lives, a half-life is 60/5 = 12 minutes.

Answer 2

Let A(t) = amount remaining after t minutes

A(t) = 312e^-kt, for some constant, k.

Since A(60) = 9.75, we have 9.75 = 312e^-60k.

So, e^-60k = 0.03125

-60k = ln(0.03125) = -3.4657

k = 0.05776

A(t) = 312e^-0.05776t

To find the half-life, set A(t) = 312/2 = 156 to get 0.5 = e^-0.05776t

So, -0.05776t = ln(0.5)

t = 12 minutes