Question

A radiation safety officer is working with 112 grams of a radioactive substance. After 17 days, the sample has decayed to 80 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance?

Answer 1

Consider that a radiation safety officer is working with 112 grams of a radioactive substance. After 17 days, the sample has decayed to 80 grams.

 

Suppose that amount of the substance remaining after t days is,

A(t) = 112at

 

Hence,

    80 = 112(a)17

  a17 = 80/112

    17 = loga(80/112)

    17 = ln(180/112)/ln(a)

ln(a) = ln(80/112)/17

ln(a) = -0.02

     a = 0.98

 

Hence, the amount of the substance remaining after t days is,

 

A(t) = 112(0.98)t

To find the half-life, put

112(0.98)t = 56

         0.98t = 1/2

                 t = log0.98(1/2)

                   = ln(1/2)/ln(0.98)

                   = 34.1

                   ≈ 34

 

Hence, the half-life of the substance is approximately 34 days.

Hence, the half-life of the substance is approximately 34 days.