accidently some radioactive material (half-life 10 days) spreads in a room. tests show that the level of radiation is 32 times the permissible level. after how many days can the room be safe for living

Let permissible no. of radioactive nuclei = N₀
Present no. of radioactive nuclei = 32N₀
Here
T_1/2  = 10 days
λ = 1/T_1/2 = 0.1
Let t = no. of days required for the room to be safe
By law of radioactive disintegration.
N₀ = 32N₀e^-0.1t
Taking ln on both sides
=> -0.1t  = ln 2^-5
=> t/10 = 5ln 2
=> t = 10 × 5× 2.303×log2
=> t = 35 days approx.