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 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 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 years to decay to one eighth of its original amount.