Ineed with this equation! carbon-14 has a half-life of 5730 years. derek has a sample of a fossil that has 33mg of its original 100 mg of carbon-14. find the decay constant, k, and use it to answer the questions that follow. first use the formula p(t)=ae ^{kt} , to find the decay constant, k

First all, the decay formula is 
where:
is the remaining quantity after  years
is the initial sample
is the time in years 
is the decay constant 

From the problem we know that  and , but we don't have the time ; to find it we will take advantage of the half-life of the Carbon-14. If you have a sample of 100 mg and Carbon-14 has a half-life of 5730, after 5730 years you will have half of your original sample i.e. 50 mg. We also know that after  years we have a remaining sample of 33mg, so the amount of the sample that decayed is . Knowing all of this we can set up a rule 3 and solve it to find :






Now that we know our time  lets replace all the values into our decay formula:

Notice that the constant  we need to find is the exponent; we must use logarithms to bring it down, but first lets isolate the exponential expression:







We can conclude that the decay constant  is approximately  -0.000144