Question

Situation:

Answer 1

Answer:

t = 6162 years

Step-by-step explanation:

This equation takes the form of

$N=N_{0}e^{kt}$

We are given everything but the amount of C-14 at time t = 0.  But we can figure it out from the info we ARE given.  We are told that when the spear head is found it contains 54% of its original amount of C-14.  Notice we are dealing in percents.  If 54% remains when it is found, it started out with 100% of its amount.  That's our N value.  Filling in:

$100=54e^{.0001t}$

Our goal is to get that t down from the exponential position that it is currently in.  To do that we will need to eventually take the natural log of both sides, because ln's and e's "undo" each other, much like squaring "undoes" a square, or dividing "undoes" multiplication.  So we take the natural log of both sides.  On the right side, notice that when we take the natural log, the e disappears; it's "undone", gone.  Before that, though, we will simplify by dividing both sides by 54.  100/54 = 1.851851852.  So, altogether...

$ln(1.851851852)=.0001t$

Simplifying by plugging the log of that number into our calculator, we get

.6161861395 = .0001t  and

t = 6161.8613

Rounding to the nearest year, t = 6162 years