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).

a)

b)

Step-by-step explanation:

For this case we have the following info provided:

represent the amount of particular radioactive isotope at time t

represent the time

represent the decay rate parameter.

Part a

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

If we reorder the expression we got:

If we integrate both sides we got:

And if we apply exponentials we got:

So then if we can rewrite the model like this:

Part b

For this case since we know that if we replace this condition in our formula we got:

So then and our model is given by:

a

explanation: