The half-life of (carbon-14) is 5730 years. that is, it takes this many years for half of a sample of to decay. if the decay of is modeled by x' = -rx, where x is the amount of , find the decay constant r. ( r = 0.000121 yr⁻¹). in an artifact the percentage of the original remaining at the present day was measured to be 20 %. how old is the artifact?

a)

We can cancel and we got:

We apply natural log and we got:

And

b)

We can cancel in both sides and we got:

Now we can apply natural log on both sides and we got:

And if we solve for t we got:

So in order to have 20% of the original amount the total time is 13301.14 years approximately.

Step-by-step explanation:

Part a

For this case we have the following model given by the differential equation:

The solution for this model is given by:

Using the half life we know that for 5730 years we need to have 1/2 of the initial amount so if we replace we have this:

We can cancel and we got:

We apply natural log and we got:

And

Where represent the initial amount. For this case we know the value for the rate of decay

Part b

And the half like is 5730 years. And we want to find the time in years in order to have 20% of the original amount. So we can write the following expression:

We can cancel in both sides and we got:

Now we can apply natural log on both sides and we got:

And if we solve for t we got:

So in order to have 20% of the original amount the total time is 13301.14 years approximately.

c

step-by-step explanation: