How old is a rock that contains 83.2% of the amount of uranium-238 it contained when it was formed?

Verified step by step guidance

1. <insert step 1> Determine the decay constant (\( \lambda \)) for uranium-238 using its half-life. The half-life of uranium-238 is approximately 4.5 billion years. Use the formula \( \lambda = \frac{0.693}{\text{half-life}} \).>
2. <insert step 2> Use the decay formula \( N = N_0 e^{-\lambda t} \), where \( N \) is the remaining quantity of uranium-238, \( N_0 \) is the initial quantity, \( \lambda \) is the decay constant, and \( t \) is the time elapsed.>
3. <insert step 3> Since the rock contains 83.2% of the original uranium-238, set \( N = 0.832 N_0 \). Substitute this into the decay formula to get \( 0.832 N_0 = N_0 e^{-\lambda t} \).>
4. <insert step 4> Simplify the equation to \( 0.832 = e^{-\lambda t} \).>
5. <insert step 5> Solve for \( t \) by taking the natural logarithm of both sides: \( \ln(0.832) = -\lambda t \). Rearrange to find \( t = \frac{\ln(0.832)}{-\lambda} \).>