What is the half-life period of a radioactive material if its...

Activity of a radioactive material is given by $A = {A}_{0}{e}^{- \lambdat}$, here, $\lambda$- is decay constant.

Taking logarithmic function both sides,

$⇒ - \lambdat = \ln \left(\dfrac{A}{{A}_{0}}\right)$

Using relation between half life and decay constant,

$⇒ - \dfrac{\ln 2}{{t}_{\dfrac{1}{2}}} \times 30 = \ln \left(\dfrac{1}{16}\right)$

$⇒ - \dfrac{\ln 2}{{t}_{\dfrac{1}{2}}} \times 30 = - 4\ln 2$

$⇒ t\dfrac{1}{2} = \dfrac{30}{4} = 7 . 5yrs$