For a radioactive material, the half-life is 10minutes . If...

Given,

${N}_{0} = 600$; ${N}^{′} = 450$

$N = {N}_{0} - {N}^{′}$

$N = 150$

Number of remaining nuclei,
$\dfrac{N}{{N}_{0}} = \left(\dfrac{1}{2}\right)\dfrac{t}{{t}_{1/2}}$ (Here, when, $t = t\dfrac{1}{2} ⇒ N = \dfrac{{N}_{0}}{2}$)

$⇒ \dfrac{150}{600} = \left(\dfrac{1}{2}\right)\dfrac{t}{10}$

$⇒ \dfrac{1}{4} = {\left(\dfrac{1}{2}\right)}^{\dfrac{t}{10}} ⇒ {\left(\dfrac{1}{2}\right)}^{2} = {\left(\dfrac{1}{2}\right)}^{\dfrac{t}{10}}$

$2 = \dfrac{t}{10} ⇒ t = 20minutes$