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

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

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7 months 2021-10-07T09:04:21+00:00 2 Answers 0 views 0

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.