Question

The first half of this problem is complete, but I dont understand how to do the second part.

Answer 1

Okay, here we have this:

According with the provided info we obtain the following equation:

$v=30\cdot(\frac{1}{2})^{\frac{t}{36}}$

Let's calculate first the 20% of the initial leval:

$30\cdot\frac{20}{100}=\frac{600}{100}=6mg$

Now, let's replace in the equation "v" with 6 to find the estimated time:

$\begin{gathered} v=30\cdot(\frac{1}{2})^{\frac{t}{36}} \\ 6=30\cdot(\frac{1}{2})^{\frac{t}{36}} \end{gathered}$

And, finally let's clear t:

$\begin{gathered} 6=30\cdot(\frac{1}{2})^{\frac{t}{36}} \\ \frac{6}{30}=(\frac{1}{2})^{\frac{t}{36}} \\ \frac{1}{5}=(\frac{1}{2})^{\frac{t}{36}} \\ \frac{t}{36}\ln (\frac{1}{2})=\ln (\frac{1}{5}) \\ t=\frac{36\ln\left(5\right)}{\ln\left(2\right)} \\ t=83.589 \\ t\approx84 \end{gathered}$

Finally we obtain that after approximately 84 hours the valium concentration will reach 20% of it's initial level.