The half-life of potassium- is billion years.

a.

Find the value of and the equation of decay.

Let is a intial amount of substance, then the amount y that remains after billion years can be represents the .

                                     (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 )

Thus, the equation for the decay of potassium- is .

b.

Find the how long will take the speciman to decay to only milligrams of potassium-.

Here and milligrams of potassium-.

        (Substitute and )

   (Divide each side by )

         (Cancel common terms)

   (Take ln on each side)

()

   (Substitute: )

   (Divide each side by )

         (Cancel common terms)

It will take the specimen about years to decay to only milligrams of Potassium-.

c.

Find  how many milligrams of potassium- will be left after million years.

Here milligrams of potassium- and  years.

        (Substitute and )

                                    (Multiply exponents)

                              (Use caluculator: )

                                            (Simplify)

About milligrams of Potassium- will be left after million years.

d.

Find how long will take Potassium- to decay to one eight of its original amount.

              (Substitute )

         (Divide each side by )

                (Cancel common terms)

  (Take ln on each side)

      ()

   (Substitute: )

(Divide each side by )

                                (Cancel common terms)

It will take Potassium- years to decay to one eighth of its original amount. \ \

a. The equation for the decay of potassium- is .

b. It will take the specimen about years to decay to only milligrams of Potassium-.

c. milligrams of Potassium- will be left after million years.

d. It will take Potassium- years to decay to one eighth of its original amount. \ \