Differential Equations: Potassium-40 Decay: The chemical element potassium is a soft metal that can

Potassium-40 Decay
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_1 K
dA/dt=λ_2 K
dK/dt=-(λ_1+λ_2)K
where λ₁ and λ₂ are positive constants of proportionality. By proceeding as in Problem 1, we can solve the mathematical model described.
(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.

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