Potassium-40 Decay One of the most abundant metals found throughout the Earth’s crust an...

Potassium-40 Decay One of the most abundant metals found throughout the Earth’s crust and oceans is potassium. Although potassium occurs naturally in the form of three isotopes, only the isotope potassium-40 (K-40) is radioactive. This isotope is a bit unusual in that it decays by two different nuclear reactions. Over time, by emitting a beta particle, a great percentage of an initial amount of K-40 decays into the stable isotope calcium- 40 (Ca-40), whereas by electron capture a smaller percentage of K-40 decays into the stable isotope argon-40 (Ar-40).* Because the rates at which the amounts C(t) of Ca-40 and A(t) of Ar-40 increase are proportional to the amount K(t) of potassium present, and the rate at which potassium decreases is also proportional to K(t) we obtain the system of linear first-order equations

where and are positive constants of proportionality.

(a) From the foregoing system of differential equations find Then find if and

(b) It is known that and Find the half-life of K-40.

(c) Use your solutions for C(t) and to determine the percentage of an initial amount of K-40 that decays into Ca-40 and the percentage that decays into Ar-40 over a very long period of time.