Radioactive technetium-99m is often used in diagnostic medicine because it has relatively short half-life but lasts long enough to get the needed testing done on the patient. If its half-life is 6 hours, how much of the radioactive material from a 2.5ml injection will be in the body in 30 hours? Write your answer rounded to nearest hundredth. This problem requires two main steps. First, we must find the decay constant k. If we start with 2.5ml, at the half-life there will be 1.25ml remaining. We will use this information to find k. Then, use that value of k to help us find the amount of sample that will be left in 30 hours.
Identify the variables in the formula.
AA0ktA=1.25=2.5=?=6hours=A0ekt
Substitute the values in the formula.
1.25=2.5ek⋅6
Solve for k. Divide each side by 2.5.
0.5=e6k
Take the natural log of each side.
ln0.5=lne6k
Use the power property.
ln0.5=6klne
Simplify.
ln0.5=6k
Divide each side by 6.

ln0.56=k (exact answer)

We use this rate of growth to predict the amount that will be left in 30 hours.
AA0ktA=?=2.5=ln0.56=30hours=A0ekt Substitute in the values.
A=2.5eln0.56⋅30
Evaluate.

A≈0.078ml

In 30 hours, there will be approximately 0.08ml remaining.