Question

A 14 gram sample of a substance thats used to sterilize surgical instruments has a k-value of 0.1481. find the substance half-life, in days round answer to the nearest tenth

Answer 1

Answer:

The half life of the substance is $\tau = 4.7 \:days$.

Step-by-step explanation:

The equation that models the amount of substance after time $t$ is

$A = A _0 e^{-kt}$.

We are told that that the initial amount $A_0= 14g$, and the k-value is $k = 0.1481$; therefore,

$A = 14e^{-0.1481t}$

The half-life of the substance is the amount of time $\tau$ it takes to decay to half its initial value; therefore,

$\dfrac{A_0}{2} = A_0e^{-0.1481\tau }$

$e^{-0.1481\tau } = \dfrac{1}{2}.$

Take the Natural Logarithm of both sides and get:

$ln[e^{-0.1481\tau } ]= ln[\dfrac{1}{2}]$

$-0.1481\tau = ln[\dfrac{1}{2} ]$

$\tau = \dfrac{ln[\dfrac{1}{2} ]}{-0.1481}$

$\boxed{\tau = 4.7 \:days}$

Thus, we find that the half life of the substance is 4.7 days.