Is this a typo on MIT OCW 18.01, problem set 2, 1H 1.b)?

Question

This question is from MIT OCW 18.01 on Single Variable Calculus from problem set 2, 1H, 1.b).

It's a 2 part question about finding the half life:

The answers are given here. However, I don't understand why we have $-\frac{\ln 2}{k}$, which I've highlighted in green. Is this a typo?

Answer 1

The problem is not strictly typographical in nature. Rather, there is an inconsistency in how the exponential decay model is specified between parts (a) and (b). In the former, the model is $$y = y_0 e^{-kt},$$ but in the latter, it is $$y = y_0 e^{kt}.$$ Both are acceptable models, but the difference is that in the former, $k > 0$ corresponds to exponential decay (and $k < 0$ corresponds to exponential growth); whereas in the latter, this relationship is reversed, so $k < 0$ corresponds to decay and $k > 0$ to growth. That is why the half-life formula is $$\lambda = \frac{\ln 2}{k}$$ in the first part, but then in the second, they use $$\lambda = -\frac{\ln 2}{k}.$$ What most likely happened here is that whoever wrote the question might have initially defined the model one way, later realized it could be clearer if it was written the other way, but failed to change both parts of the solution.

Answer 2

You probably realized this by now, but there are actually two typos in (b). The first is $y_1=y_0e^{kt_1}$. It should be $y_1=y_0e^{-kt_1}$. The second is $\lambda=-\ln 2 / k$; this should be $\lambda = \ln 2/k$.