Assuming 1μ g of trace radioactive element X with a half life of 30...

$\lambda t = \ln \left(\dfrac{a}{a - x}\right)$

we know that half life time ${t}_{1/2} = \dfrac{\ln 2}{\lambda}$

So, $\lambda = \dfrac{\ln 2}{{t}_{1/2}}$

$100 = \dfrac{30}{\ln 2}\ln \left(\dfrac{1}{w}\right)100 = \dfrac{30}{\log 2}\log \left(\dfrac{1}{w}\right)100 = \dfrac{30}{0 . 30}\log \left(\dfrac{1}{w}\right)\log \left(\dfrac{1}{w}\right) = 1$

$\dfrac{1}{w} = 10$

$W = 0 . 1\mu g$

=$1 \times {10}^{- 1}\mu g$