Chapter 4.5, Problem 81E

Radioactive Decay A 15-g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by m(t) = 15e^−0.087t, where m(t) is measured in grams. After how many days are there only 5 g remaining?

To determine

To find: The number of days in which the $15g$ sample of radioactive become $5g$.

Answer to Problem 81E

The sample of $15g$ radioactive become $5g$ in $13$ days.

Explanation of Solution

Given:

The equation for the mass remaining after $t$ days is, $m\left(t\right)=15e^{-0.087t}$.

Remaining radioactive iodine is $5g$.

Calculation:

The given equation for the mass remaining after $t$ days is,

$m\left(t\right)=15e^{-0.087t}$

Substitute $5$ for $m\left(t\right)$,

$\begin{array}{c}5=15e^{-0.087t}\\ \frac{1}{3}=e^{-0.087t}\\ \ln \left(\frac{1}{3}\right)=\ln e^{-0.087t}\left(\because \text{Taking}\text{ln}\text{each}\text{side}\right)\\ -1.09861=-0.087t\left(\because \ln e^x=x\right)\end{array}$

Further simplify the above expression,

$\begin{array}{c}t=\frac{1.09861}{0.087}\\ t=12.6277\\ r\approx 13\end{array}$

Thus, the sample of $15g$ radioactive become $5g$ in $13$ days.