Overview:
To determine the age of a fossil, we can use the concept of half-life. The half-life is the time it takes for half of the radioactive material to decay. By measuring the ratio of radioactive isotope to stable isotope in a sample, we can calculate the number of half-lives that have passed and hence estimate the age of the fossil.

Given information:
- Rock sediment contains 100 gm of radioactive carbon C14 and 700 gm of stable isotope N14 in a sample from the same rock.

Step 1: Calculate the initial ratio:
The initial ratio of C14 to N14 in the sample can be calculated by dividing the mass of C14 by the mass of N14:
Initial ratio = 100 gm / 700 gm = 1/7

Step 2: Determine the half-life of C14:
The half-life of C14 is known to be approximately 5730 years.

Step 3: Calculate the number of half-lives:
To determine the number of half-lives that have passed, we need to compare the current ratio of C14 to N14 with the initial ratio.

Step 3.1: Calculate the current ratio:
Let's assume the current ratio of C14 to N14 is x. We can set up the following equation based on the decay of C14 over time:
x = (1/2)^n

Where n is the number of half-lives that have passed.

Step 3.2: Solve for n:
We can rewrite the equation as:
log(x) = log((1/2)^n)
log(x) = n * log(1/2)
n = log(x) / log(1/2)

Step 3.3: Calculate x:
To calculate x, we can divide the mass of C14 by the mass of N14 in the current sample:
x = 100 gm / 700 gm = 1/7

Step 3.4: Calculate the number of half-lives:
Substituting the value of x into the equation:
n = log(1/7) / log(1/2)
n ≈ 3.8

Therefore, approximately 3.8 half-lives have passed.

Step 4: Calculate the age of the fossil:
To calculate the age of the fossil, we can multiply the number of half-lives by the half-life of C14:
Age = 3.8 * 5730 years
Age ≈ 21774 years

Conclusion:
The age of the fossil recovered from the rock sediment is estimated to be approximately 21774 years. By using the initial ratio of C14 to N14, the known half-life of C14, and the current ratio of C14 to N14, we were able to calculate the number of half-lives that have passed and hence estimate the age of the fossil.

Options are..... (a) 2 HL(b) 3 HL(c) 5HL (d) 4HL