Please provide solution of this problem

Assuming all the disintegrations are detected by the counter.

count rate = activity

dN/dt = -λe^-λt N₀

16000 = -λe^-λt N₀ .....(1)

After 4 hrs i.e. 240 min,

500 = -λe^-λ(t+240) N₀ ....(2)

Divide (2) by (1),

500/16000 = e^{+λt-λ(t+240)}

500/16000 = e^-240λ

ln (1/32) = -240λ

Use t_1/2 = ln2/λ

ln (1/32) = -240(ln2/t_1/2)

t_1/2 = 240 ln2 / ln32

t_1/2 = 240 ln2 / ln32 = 48 min.

Half life of the radioactive material is 48 min.

Regards,

Team,

TopperLearning.