A piece of bone of an animal from a ruin is found to have 14C activity of 12 disintegrations per minute per gm of its carbon content. The 14C activity of a living animal is 16 disintegrations per minute per gm. Nearly, how long ago did the animal die?

We know that:
A = Ao e^(-λt)
where:
A = activity of the sample at time t = 12 disintegrations per minute per gm
Ao = initial activity of the sample (activity at time t=0) = 16 disintegrations per minute per gm
λ = decay constant for 14C = 0.693/t1/2 (half-life of 14C is approximately 5730 years)
t = time elapsed since the animal died

Substituting the given values:
12 = 16 e^(-λt)
0.75 = e^(-λt)
Taking natural logarithm on both sides:
ln(0.75) = -λt
t = -ln(0.75)/λ

We know that λ = 0.693/t1/2 = 0.693/5730 years ≈ 1.21 x 10^-4 per year

Therefore:
t = -ln(0.75) / (1.21 x 10^-4 per year)
t ≈ 2391 years

Thus, the animal died approximately 2391 years ago.