Question

I don't understand the answer to 1H-1b)--how/why was the assumption made?

Answer 1

Their explanation is a bit muddled. They are saying that lambda has the value found
from part a): the half-life (given the name lambda) is \( \lambda= \frac{\ln 2}{k} \)

For part 2) we are given
\[ y_1= y_0\ e^{-k t_1} \]
now find y at time \(t_1 + \lambda\):
\[y_2 = y_0\ e^{-k (t_1+ \lambda)} \\
= y_0\ e^{-k t_1}e^{-k \lambda} \\
= y_1\ e^{-k \cdot \frac{\ln 2}{k} } \\
= y_1 \cdot \frac{1}{e^{\ln 2}} \\
y_2= \frac{y_1}{2}
\]