Answer:
4.12 billion years
Step-by-step explanation:
To determine the age of the rock sample, we can use the concept of radioactive decay. Potassium-40 (K-40) decays into argon-40 (Ar-40) with a half-life of 1.3 billion years. The ratio of Ar-40 to K-40 in a rock sample can be used to calculate the age of the sample. Here are the steps to solve this problem:
Step 1: Determine the initial ratio of Ar-40 to K-40 when the rock was formed. This is the ratio at the time the rock was created.
Step 2: Calculate the current ratio of Ar-40 to K-40 in the rock sample. This is the ratio you've provided: 5,520 atoms of Ar-40 and 480 atoms of K-40.
Step 3: Calculate the number of half-lives that have passed since the rock was formed. To do this, use the formula:
Number of Half-Lives = (ln(Current Ratio / Initial Ratio)) / (ln(1/2))
Where ln represents the natural logarithm.
Step 4: Calculate the age of the rock sample using the half-life and the number of half-lives:
Age = Number of Half-Lives x Half-Life
Step 5: Plug in the values from your calculations:
Age = (Number of Half-Lives) x 1.3 billion years
Substituting the values you provided:
Step 1: The initial ratio of Ar-40 to K-40 is not given in the question, so we'll proceed with the provided information.
Step 2: The current ratio is 5,520 atoms of Ar-40 to 480 atoms of K-40.
Step 3: First, calculate the ratio:
Ratio = (5520 / 480) = 11.5
Now, calculate the number of half-lives:
Number of Half-Lives = (ln(11.5)) / (ln(1/2)) ≈ 3.16 (rounded to 2 decimal places)
Step 4: Calculate the age:
Age = 3.16 x 1.3 billion years ≈ 4.12 billion years (rounded to 2 decimal places)
So, the age of the rock sample is approximately 4.12 billion years.