A storage tank contains a liquid radioactive element with a half-life of 96 years. 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?
The tank must remain intact for years.
(Round the base of the exponential function to four decimal places. Then round the final answer to the nearest year as needed.)

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.

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