Final answer:
The age of a rock with a mass ratio of argon-40 to potassium-40 of 4.2, and a half-life of potassium-40 of 1.27×109 years, is calculated to be approximately 1.7 billion years old using the potassium-argon dating method.
Step-by-step explanation:
Calculating Rock Age Using Potassium-Argon Dating
The question asks for the age of a rock, given that the mass ratio of argon-40 (40Ar) to potassium-40 (40K) is 4.2. To find the age of the rock using the potassium-argon dating method, we can apply the concept that 40K decays into 40Ar with a known half-life. This half-life is given as 1.27×109 years.
Using the given mass ratio of 40Ar to 40K, which is 4.2, we can estimate the number of half-lives that have passed. Each half-life passed will result in the remaining 40K being half of what it was before and an increase in 40Ar. We can express the ratio in terms of the decay equation:
NAr / NK = (e−0.693t/T) / (1 − e−0.693t/T)
Where NAr is the amount of 40Ar, NK is the amount of remaining 40K, t is the age of the rock, and T is the half-life of 40K. Solving for t using the given ratio and half-life, we find that the rock is approximately 1.7 billion years old.