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Every chemical element goes through natural exponential decay, which means that over time its atoms fall apart. The speed of each element's decay is described by its half-life, which is the amount of time it takes for the number of radioactive atoms of this element to be reduced by half.

The half-life of the isotope beryllium-11 is 14 seconds. A sample of beryllium-11 was first measured to have 800 atoms. After t seconds, there were only 50 atoms of this isotope remaining.

Requried:
Write an equation in terms of t that models the situation.

Respuesta :

Answer:

[tex]Q(t) = 800e^{-0.0495t}[/tex]

Step-by-step explanation:

The amount of the isotope after t seconds is given by the following exponential equation:

[tex]Q(t) = Q(0)e^{-rt}[/tex]

In which Q(0) is the initial amount and r is the decay rate.

A sample of beryllium-11 was first measured to have 800 atoms.

This means that [tex]Q(0) = 800[/tex]

The half-life of the isotope beryllium-11 is 14 seconds.

This means that [tex]Q(14) = 0.5Q(0)[/tex]

We use this to find r.

[tex]Q(t) = Q(0)e^{-rt}[/tex]

[tex]0.5Q(0) = Q(0)e^{-14r}[/tex]

[tex]e^{-14r} = 0.5[/tex]

[tex]\ln{e^{-14r}} = \ln{0.5}[/tex]

[tex]-14r = \ln{0.5}[/tex]

[tex]r = -\frac{\ln{0.5}}{14}[/tex]

[tex]r = 0.0495[/tex]

So

[tex]Q(t) = Q(0)e^{-rt}[/tex]

[tex]Q(t) = 800e^{-0.0495t}[/tex]