The chemical element potassium is a soft metal that can be found extensively throughout the Earth’s crust and oceans. Although potassium occurs naturally in the form of three isotopes, only the isotope potassium-40 (K-40) is radioactive. This isotope is also unusual in that it decays by two different nuclear reactions. Over time, by emitting beta particles a great percentage of an initial amount K₀ of K-40 decays into the stable isotope calcium-40 (Ca-40), whereas by electron capture a smaller percentage of K₀ 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 K(t) decays is also proportional to K(t), we obtain the system of linear fi­rst-order equations

dC/dt = λ_­1K

dA/dt = λ₂K

dK/dt = -(λ₁ + λ₂)K,

where λ₁ and λ₂ are positive constants of proportionality. By proceeding as in Problem 1 we can solve the foregoing mathematical model.

(a) From the last equation in the given system of differential equations fi­nd K(t) if K(0) = K₀. Then use K(t) to fi­nd C(t) and A(t) from the fi­rst and second equations. Assume that C(0) = 0 and A(0) = 0.

(b) It is known that λ₁ = 4.7526 × 10^-10 and λ₂ = 0.5874 × 10^-10. Find the half-life of K-40.

(c) Use C(t) and A(t) found in part (a) to determine the percentage of an initial amount K₀ of K- 40 that decays into Ca-40 and the percentage that decays into Ar-40 over a very long period of time.