Question

Situation:A 34 gram sample of a substance that's used to treat thyroid disorders has a k-value of 0.137.N= Noe-ktNo initial mass (at time t = 0)N = mass at time tk== a positive constant that depends on the substance itself and on the units used to measure timet = time, in daysFind the substance's half-life, in days.Round your answer to the nearest tenth.

Answer 1

Answer:

5.1 days

Explanation:

The half-life of a substance is the time it takes the substance to decay to half of its initial mass.

The function that models the decay of the substance is:

$N=N_oe^{-kt}$

• The initial mass, No = 34 grams

,

• Half of the initial mass, N = 34/2 = 17 grams

,

• k=0.137

Substitute these values into the formula:

$17=34e^{-0.137t}$

The equation is then solved for t:

$\begin{gathered} \text{ Divide both sides by 34} \\ \frac{17}{34}=\frac{34e^{-0.137t}}{34} \\ e^{-0.137t}=0.5 \\ \text{ Take the }\ln\text{ of both sides} \\ \ln(e^{-0.137t})=\ln(0.5) \\ -0.137t=\ln(0.5) \\ \text{ Divide both sides by -0.137} \\ \frac{-0.137t}{-0.137}=\frac{\operatorname{\ln}(0.5)}{-0.137} \\ t=5.06 \\ t\approx5.1\text{ days} \end{gathered}$

The substance's half-life is 5.1 days (rounded to the nearest tenth).