a half life of a certain radioactive material is 10 hours and initial amount of the material has a mass of 75 Kg. find how long it will take until there is 15 kg of the radioactive material remaining
we know that
the equation that represent this situation is an exponential function of the form
[tex]y=a(\frac{1}{2})^t[/tex]where
a is the initial value
t is the time in hours
t=time/half life
y is the mass
In this problem we have
a=75 kg
t=t/10
so
substitute
[tex]y=75(\frac{1}{2})^{(\frac{t}{10)})}[/tex]For y=15 kg
substitute in the equation above
[tex]\begin{gathered} 15=75(\frac{1}{2})^{(\frac{t}{10)})} \\ \frac{15}{75}=(0.5)^{(\frac{t}{10})} \\ \text{apply log both sides} \\ \log (\frac{15}{75})=\frac{t}{10}\cdot\log (0.5) \\ t=10\cdot\log (\frac{15}{75})\colon\log (0.5) \\ t=23.2\text{ hours} \end{gathered}[/tex]therefore
