Answer:
The tank must remain intact for 1183 years.
Step-by-step explanation:
Exponential equation for decay:
The amount of a substance after t years is given by:
[tex]A(t) = A(0)e^{rt}[/tex]
In which A(0) is the initial amount and r is the decay rate.
A storage tank contains a liquid radioactive element with a half-life of 96 years.
This means that [tex]A(96) = 0.5A(0)[/tex], and we use this to find r.
[tex]A(t) = A(0)e^{rt}[/tex]
[tex]0.5A(0) = A(0)e^{96r}[/tex]
[tex]e^{96r} = 0.5[/tex]
[tex]\ln{e^{96r}} = \ln{0.5}[/tex]
[tex]96r = \ln{0.5}[/tex]
[tex]r = \frac{\ln{0.5}}{96}[/tex]
[tex]r = -0.0072[/tex]
So
[tex]A(t) = A(0)e^{-0.0072t}[/tex]
It will be relatively safe for the contents to leak from the tank when 0.02% of the radioactive element remains. How long must the tank remain intact for this storage procedure to be safe?
This is t for which [tex]A(t) = 0.0002A(0)[/tex]. So
[tex]A(t) = A(0)e^{-0.0072t}[/tex]
[tex]0.0002A(0) = A(0)e^{-0.0072t}[/tex]
[tex]e^{-0.0072t} = 0.0002[/tex]
[tex]\ln{e^{-0.0072t}} = \ln{0.0002}[/tex]
[tex]-0.0072t = \ln{0.0002}[/tex]
[tex]t = -\frac{\ln{0.0002}}{0.0072}[/tex]
[tex]t = 1183[/tex]
The tank must remain intact for 1183 years.
