Question

The radioactive isotope carbon-11 is used in medicine for brain-scan procedures. It decays losing half its mass every 20 minutes. If a doctor checks on a patient two hours after the isotope has been administered and 0.10 microgram of it still remains, what is the amount of isotope carbon-11 administered to the patient.

Answer 1

6.4 micrograms were administered to the patient.

The general form of this equation would be:

$A=p(1+r)^t$, where A is the total amount left, p is the initial amount, r is the rate of decay, and t is the number of time periods.

For our question, A =0.1 since that is the amount left; p is unknown; r is -0.5, since it loses half of its mass; and t will be 6.  2 hours = 120 minutes/20 minutes = 6 20-minute increments:

$0.10 = p(1-0.5)^6 \\ \\0.10=p(0.5)^6$

Divide both sides by 0.5^6:
$\frac{0.10}{0.5^6}=p \\ \\6.4=p$