The substance decay is   of its total mass after years.

a.

Find the half-life of substance.

The substance decay is then .

                            (Exponential decay formula)

            (Substitute and )

       (Divide each side by )

                (Cancel common terms)

    (Take ln on each side)

         ()

     (Divide each side by )

                 (Cancel common terms)

            (Substitute: )

               (Simplify)

                      (Substitute )

Let is a intial amount of substance, then the amount can be represents the .

                      (Exponential decay at )

                (Substitute )

           (Divide each side by )

                    (Cancel common terms)

       (Take ln on each side)

           ()

   (Substitute: )

(Divide each side by )

                             (Cancel common terms)

The half–life of the substance is about years.

The half–life of the substance is about years.