2022 AHS AM S4 P1 Q4

Carbon-14 has a half-life of 5730 years, which means that it takes 5730 years for the original mass of Carbon-14 to be reduced by 50%. To determine the age of a plant or animal fossil, scientists determine the amount of Carbon-14 in the specimen as the Carbon-14 undergoes radioactive decay.

The amount of Carbon-14 in a piece of fossilised bone is given by \(M = M_0e^{-kt}\), where \(k\) is a constant, \(M\) is the mass of Carbon-14 in the specimen, \(M_0\) is the initial mass of Carbon-14 and \(t\) is measured in years.

1. Determine the value of \(k\).[2]
2. Determine the percentage of Carbon-14 that has decayed in the specimen if the fossil is estimated to be about 900 years old.[2]
3. Express the rate of change of percentage of Carbon-14 in terms of \(t\).[2]