Most drugs in the bloodstream decay by $y' = cy$ (first-order kinetics). (a) The half-life of morphine is 3 hours. Find its decay constant $c$ (with units). (b) The half-life of nicotine is 2 hours. After a six-hour flight what fraction remains?
For (a), $c=-\frac{\ln2}3\approx-0.2310$. Should the unit be "per hour"? It is curious though that the key does not include the unit.
For (b), $c=-\frac{\ln2}2$ and $\frac y{y_0}=e^{ct}=e^{-\frac{\ln2}2\cdot6}=e^{-3\ln2}=\frac1{2e^3}\approx0.0249$. The key says $\left(\frac12\right)^3=\frac18$. Where was I wrong?
For (a), yes the units of the constant $c$ are "per hour". Answer keys for calculus problems frequently leave off units.
For (b), the error is in simplifying $e^{-3\ln 2}$.
Using the property that $x^{ab} = (x^b)^a$ for $x > 0$, we have $e^{-3 \ln 2} = (e^{\ln 2})^{-3} = 2^{-3} = \dfrac{1}{8}$.