As you know, radioactive decay is an exponetial decay.

The equation for exponential decay is given by,

`(dN)/(dt) = -lambdaN`

where N - Number of activity or amount at time t and `lambda` is decay constant.

Rearranging thae above equation you get,

`(dN)/N = -lambdadt`

Integrating this gives,

`ln(N) = -lambdat+C`

where C is a constant.

Now we have to find `lambda` and C by using the given data.

at t = 0, N = 500, then,

ln(500) = C

therefore the equation changes to,

`ln(N) = -lambdat+ln(500)`

`ln(N/500) = -lambdat`

The half-time of this is 139 days, That means at 139 days N will be 250 mili grams.

`ln(250/500) = -lambda * 139`

`ln(1/2) = -lambda * 139`

`ln(2) = lambda *139`

`lambda = ln(2)/139 = 0.6931/139`

therefore the equation changes to,

`ln(N/500) = -(0.6931/139)t`

a) How many left after 30 days??

`ln(N/500) = -(0.6931/139)*30`

`ln(N/500) = -0.1496`

`N/500 = e^(-0.1496)`

`N/500 = 0.861`

N = 430.5

**Therefore after 30 days 430.5 mg will be left.**

b)How many days to get to 100 mg,

`ln(100/500) = - (0.6931/139) * t`

`ln(1/5) = -(0.6931/139) *t`

-1.6094 =-(0.6931/139) *t

t = 322.76 days.

**The days required to pass to have only 100 mg is 322.76 days.**