Answer:
The half life of the substance is [tex]\tau = 4.7 \:days[/tex].
Step-by-step explanation:
The equation that models the amount of substance after time [tex]t[/tex] is
[tex]A = A _0 e^{-kt}[/tex].
We are told that that the initial amount [tex]A_0= 14g[/tex], and the k-value is [tex]k = 0.1481[/tex]; therefore,
[tex]A = 14e^{-0.1481t}[/tex]
The half-life of the substance is the amount of time [tex]\tau[/tex] it takes to decay to half its initial value; therefore,
[tex]\dfrac{A_0}{2} = A_0e^{-0.1481\tau }[/tex]
[tex]e^{-0.1481\tau } = \dfrac{1}{2}.[/tex]
Take the Natural Logarithm of both sides and get:
[tex]ln[e^{-0.1481\tau } ]= ln[\dfrac{1}{2}][/tex]
[tex]-0.1481\tau = ln[\dfrac{1}{2} ][/tex]
[tex]\tau = \dfrac{ln[\dfrac{1}{2} ]}{-0.1481}[/tex]
[tex]\boxed{\tau = 4.7 \:days}[/tex]
Thus, we find that the half life of the substance is 4.7 days.
