# Question 2ac43

Dec 3, 2014

I believe the answer is 108 hours.

An exponential decay process can be described by the following equation:

N(t) = N_0(t) * (!/2)^(t/t_(1/2))# , where

$N \left(t\right)$ - the initial quantity of the substance that will decay;
${N}_{0} \left(t\right)$ - the quantity that still remains and has not yet decayed after a time t;
${t}_{\frac{1}{2}}$ - the half-life of the decaying quantity;

This being said, we know that our ${t}_{\frac{1}{2}}$ is equal to 21.6 hours, our $N \left(t\right)$ is 11.25 grams, and our ${N}_{0} \left(t\right)$ is equal to 360 grams.

Therefore,

$11.25 = 360 \cdot {\left(\frac{1}{2}\right)}^{\frac{t}{21.6}}$ . Now, let's say $\frac{t}{21.6}$ is equal to $y$.
We then have

$\frac{11.25}{360} = {\left(\frac{1}{2}\right)}^{y}$

So $y = {\log}_{\frac{1}{2}} \left(0.03125\right) = 5$

Replacing this into

$\frac{t}{21.6} = 5$ we get $t = 108 h o u r s$