Radioactive isotope X has a half-life of 500,000 years. This isotope may be found in some types of volcanic rocks. A particular sample of volcanic rock taken from a layer that covered up some of the earliest known human-like footprints contains 0.125 mg of isotope X. The volcanic rock sample originally contained 8.00 mg of isotope X. How long ago were these footprints made. I need the equation. Thanks. please I am stuck on this question. PLEASE

The age of the earliest known human-like footprints is approximately 3000000 years.

Explanation:

Decay of isotopes is represented by the following ordinary differential equation:

(Eq. 1)

Where:

- First derivative of mass with respect to time, measured in miligrams per year.

- Current mass of the isotope, measured in miligrams.

- Time constant, measured in years.

Now we proceed to obtain the solution of the differential equation:

(Eq. 2)

Where:

- Initial mass of the isotope, measured in miligrams.

- Time, measured in years.

- Time constant, measured in years.

We proceed to clear time within the formula presented above:

In addition, time constant can be found as a function of half-life:

(Eq. 3)

If we know that , and , the age of the earliest known human-like footprints is:

The age of the earliest known human-like footprints is approximately 3000000 years.