A sample of a radioactive isotope is monitored over time. The table lists the mass M(t) (in grams) remaining at selected times t (in days).
t
0
5
10
15
20
25
M(t)
120
96
76
60
48
38
Statistical software fits two different linear models to transformed versions of the data:
• Exponential model fit: lnM=4.787−0.045t with R2=0.994.
• Alternative model fit: 1/M=0.0068+0.00012t with R2=0.981.
Assume that exactly one of these two models is appropriate for the isotope’s decay.
(a) For the exponential model, determine the decay constant k and write the corresponding exponential expression for M(t).
(b) Using the model in part (a), calculate the isotope’s half-life to the nearest day.
(c) Based on the given R2 values and on qualitative considerations of residual patterns (no calculations required), justify which of the two models should be selected to describe the decay. Provide at least two distinct reasons.
(d) Discuss one limitation of applying the selected model far beyond the 25-day observation window, referencing the physical context of radioactive decay.