This exercise uses the radioactive decay model. after 3 days a sample of radon-222 has decayed to 58% of its original amount. (a) what is the half-life of radon-222? (round your answer to two decimal places.) days (b) how long will it take the sample to decay to 15% of its original amount? (round your answer to two decimal places.)

a) T= 3.82 days b) t=1.05 days

Explanation:

 (a)  

The radioactive decay model is given by the mathematical expression that defines the exponential decrease, which can be expressed as:

N   =   N0 ·   0.5 t/T

Beaing

T = half life in days

t = days

N0= Number of atoms initially

N = Number of remaining atoms

 N(t) = N0 (0.5)t/T

 For the data of the problem, it is necessary to know:

N(t)/ N0 = %N/100

So

0.58 = (0.5)3/T

I apply logarithm to clear T

 log(0.58) = (3/T) log(0.5)

 T = 3 log(0.5) / log(0.58) days = 3.82 days

 (b)  

  I use the previous formulas. So I have left

 0.15 = (0.5)t/T = (0.5)t/3.82

log(0.15) = (t/3.82) log(0.5)

t=3.82log(0.15)/ log(0.15)

t=1.05 days