A radioactive nucleus X decays to a nucleus Y with a decay constant `lambda_X=0.1s^-1, Y` further decays to a stable nucleus Z with a decay constant
`lambda_Y=1//30s^-1`. Initially, there are only X nuclei and their number is `N _0=10^20`. Set up the rate equations for the populations of X, Y and Z. The population of Y nucleus as a function of time is given by `N_Y(t)={N _0lambda_X//(lambda_X-lambda_Y)}[exp(-lambda_Yt)-exp(-lambda_Xt)].`
Find the time at which `N_Y` is maximum and determine the population X and Z at that instant.