Problem 6: The amount of a certain medicine in the bloodstream decays exponentially with a half-life of 5 hours. In order to keep a patient safe during a one-hour procedure, there needs to be at least 50 mg of medicine per kg of body weight. How much medicine should be administered to a 60kg patient at the start of the procedure?
Let $x(t)$ be the amount of the medicine in mg present in the bloodstream at time $t$ in hours.
Then we have the model: $x(t) = x_0e^{-kt}$
I wonder how in the solution they found $k = \frac{\ln2}{5}$.
How I tried: We know that after $t=5$ hours, the initial amount is halved, so $x(5) = (1/2)x_0$.
Then we use that information to solve for $k$:
$$ (1/2)x_0=x_0e^{-k(5)} $$
And then:
$$ ln(1/2) = -5k $$
Etc. so that $k=-\frac{\ln(1/2)}{5}$.
However in the given solution they leave out the minus sign when finding k (which I can understand) and more importantly, they find $k = \frac{\ln2}{5}$, where I have $\ln(1/2)$ in the numerator.
$$ - \frac{\ln(1/2)}{5} = - \frac{\ln 2^{-1}}{5} = \frac{\ln (2^{-1})^{-1}}{5} = \frac{\ln 2}{5}. $$