Question

Model radioactive decay using the notation t = time (independent variable), r(t) = amount of particular radioactive isotope present at time t (dependent variable), -lambda = decay rate (parameter). (a) Using this notation, write a model for the decay of a particular radioactive isotope. (b) If the amount of the isotope present at t = 0 is r_0, state the corresponding initial-value problem for the model in part (a).

Answer 1

a)
`(dr)/(dt)= -\lambda r`

`r(t) = K e^(-\lambda t)`

b)
`r(t)= r_o e^(-\lambda t)`

Explanation:

For this case we have the following info provided:

`r(t)` represent the amount of particular radioactive isotope at time t

`t` represent the time

`-\lambda` represent the decay rate parameter.

Part a

We can use the following proportional model given by this differential equation:

`(dr)/(dt)= -\lambda r`

If we reorder the expression we got:

`(dr)/(r) = - \lambda dt`

If we integrate both sides we got:

`ln|r| = -\lambda t +C`

And if we apply exponentials we got:

`r = e^(-\lambda t) e^C`

So then if
`e^C =k`we can rewrite the model like this:

`r(t) = K e^(-\lambda t)`

Part b

For this case since we know that
`t=0, r(0) = r_o` if we replace this condition in our formula we got:

`r_o = K e^(-\lambda *0) =K`

So then
`K=r_o` and our model is given by:

`r(t)= r_o e^(-\lambda t)`