A 10 kg sample from a living plant has actiuity 30 000 Bq due to Carbon-14. A 500g sample of the same type of plant but dead has activity 1 000 Bq.

Calculate the age of the dead sample given that the half life of Carbon-14 is 5568 years

To calculate the age of the dead sample, you can use the concept of exponential decay. The decay of Carbon-14 over time can be modeled by the equation:

N(t) = N0 * (1/2)^(t/T)

Where:
N(t) is the amount of Carbon-14 remaining at time t
N0 is the initial amount of Carbon-14
t is the time that has passed
T is the half-life of Carbon-14

In your case, the living plant sample had an activity of 30,000 Bq, and the dead plant sample had an activity of 1,000 Bq. We can start by determining the initial amount of Carbon-14 in the dead sample.

N0 = N(t) * (2^(t/T))

N0 = 1000 Bq * (2^(t/5568))

Since the mass of the dead sample is 500g, the mass of the living sample would be 10kg. We can now determine the initial amount of Carbon-14 in the living sample.

N0(living) = N(living) * (2^(t/5568))

N0(living) = 30000 Bq * (2^(t/5568))

Since the two samples are of the same type of plant, the initial amount of Carbon-14 should be the same.

N0(living) = N0

30000 Bq * (2^(t/5568)) = 1000 Bq * (2^(t/5568))

We can now solve this equation to find the value of t, which represents the age of the dead sample.

Divide both sides of the equation by 1000 Bq * (2^(t/5568)) to get:

30 = 2^(t/5568)

Take the logarithm (base 2) of both sides to get:

log2(30) = t/5568

Multiply both sides by 5568 to isolate t:

t = 5568 * log2(30)

Using a scientific calculator, you can evaluate this expression to find the age, t, of the dead sample.