Question 314890


Your given equation:
{{{N=Ie^(kt)}}}
is the standard exponential growth/decay formula
where
N is the amount after time t
I is the initial amount
k is the growth/decay rate
t is time
.
If k is positive it is "growth" otherwise k is negative or "decay".
.
Given your problem:
an artifact is found at a certain site. If it has 65% of the carbon-14 it originally contained, what is the approximate age of the artifact?
(carbon-14 decays at the rate of 0.0125% annually)
.
Notice the problem doesn't give you the initial amount so assign it a variable:
Let x = initial amount
then
N is .65x
I is x
k is 0.0125% or 0.000125
t is what we need to find
.
Plug in what was given into:
{{{N=Ie^(kt)}}}
{{{ .65x = xe^(0.000125t)}}}
Now, solve for t:
Start, by dividing both sides by x:
{{{ .65 = e^(0.000125t)}}}
Next, take the natural log of both sides:
{{{ ln(.65) = 0.000125t }}}
{{{ ln(.65)/0.000125 = t }}}
{{{ -3446.26 = t }}} years
.
Or, it is about 3446 years old.