Question on Decay Rate?

Question

A 12-g sample of carbon from living matter decays at the rate of 180 decays/min due to the radioactive carbon 14 in it. What will be the decay rate of this sample in a 1000 years? The half-life of carbon 14 is 5730 years. The answer is 159 decays/min.

I don't understand why they are not converting 180 decays/min in decays/year when they solved this question.

Answer 1

what is left after 1000 years based off of the 14 rate, because I am going 80 mph now does not mean in 1000 years i will be going 80 mph.

Answer 2

lambda * t  = 2.303 log (Ao / A) ......... (1)

here, lambda = decay constant

t = time

Ao = Original activity

A = Activity at time t

also, lambda = 0.6931/ T ( T = Half life) ........ (2)

Putting (2) and given values in  (1):

0.6931 * 1000 / 5730 = 2.303 log (180/A)

{ Actually, ideally we shd have converted the activity 180 decays/min into decays/year because the value of T and hence lambda is in terms of year not mins , but with this formula the conversion terms will cancel from both numerator and denominator, so u still get the right answer. But, if they mark u for steps in ur xam u will have to be careful.}

solving, u get :

log (180/A) = 0.05252

=> ( log 180 - log A) = 0.05252

log 180 = 2.2552

so, solving, log A = 2.202752 (phew!)

or, A = antilog 2.202752 i.e A = 159.6 decays / min

That was LONG!