Q.The rate of disintegration of a radioactive element at any time $t$ is proportional to its mass at that time. Find the time during which the original mass of 1.5 gm will disintegrate into its mass of 0.5 gm.

Let $x=x_0e^{-kt}$ with $x_0=1.5$ gm. Disintegrating to $0.5$ gm: $0.5=1.5\,e^{-kt}\Rightarrow e^{-kt}=\dfrac13\Rightarrow -kt=-\log3\Rightarrow t=\dfrac{\log3}{k}$ — expressed in terms of the (unspecified) decay constant $k$, s …

Model the situation as dx/dt proportional to x (or to (θ-θ0) for cooling), solve the resulting first-order linear/separable equation to get x = c·e^{kt} (or the Newton's-law form), then use the gi …

Mixing up which measured instant corresponds to t (measuring time from the WRONG reference point); forgetting that a ratio like x(t2)/x(t1)=e^{k(t2-t1)} lets k cancel out algebraically, and instead trying …