A radio nuclide X is produced at constant rate `alpha`. At time `t=0`, number of nuclei of X are zero. Find
(a) the maximum number of nuclei of X.
(b) the number of nuclei at time t.
Decay constant of X is `lambda`.

Explanation

(a) `(dN)/(dt) = alpha - lambdaN`
`underset(0)overset(N)int (dN)/((alpha - lambdaN)) =underset(0)overset(t)int dt`
`(|ln(alpha - lambdaN)|_(0)^(N))/(-lambda)=t`
`ln(alpha - lambdaN)-ln(alpha) = -lambdat`
`ln ((alpha - alphaN)/(alpha)) =- lambdat`
`(alpha -lambdaN)/(alpha) = e^(-lambdat)`
`N = (alpha)/(lambda)[1-e^(-lambdat)]`
(b) When `t rarr oo, e^(-lambdat) rarr 0`
`N = (alpha)/(lambda)`