An artifact was found and tested for its carbon-14 content. If 88% of the original carbon-14 was still present, what is its probable age (to the nearest 100 years)? Use that carbon-14 has a half-life of 5,730 years.

you could use

amount = initial(1/2)^(t/5730)
.88 = 1(1/2)^(t/5730)
t/5730ln.5 = ln .88
t = 5730ln.88/ln.5 = 1056.75 yrs.

or

let amount = initial (e^(kt))
when t = 5730 amount = .5
.5 = e^(5730k)
k = ln.5/5730 = -0.000120968

.88 = e^(-.000120968t)
t = ln.88/-.000120968
= 1056.75

To find the probable age of the artifact, we need to use the concept of exponential decay and the half-life of carbon-14.

The half-life of carbon-14 is 5,730 years, which means that after a period of 5,730 years, half of the carbon-14 atoms in a sample will have decayed. We can use this information to create an equation relating the remaining percentage of carbon-14 to the number of half-lives.

Let's use N to represent the number of half-lives. We know that 88% of the original carbon-14 is still present, so we have 88% as a decimal, which is 0.88. Therefore, the equation we can form is:

Remaining Carbon-14 = Original Carbon-14 * (1/2)^N

where N is the number of half-lives.

We need to solve for N. To do so, we can take the logarithm (base 2) of both sides of the equation to remove the exponential term:

log2(Remaining Carbon-14 / Original Carbon-14) = N

where log2 represents the logarithm to the base 2.

Now we can substitute the given value of Remaining Carbon-14, which is 0.88, and solve for N:

log2(0.88 / 1) = N

Using a calculator, we find that N is approximately -0.126. Since we cannot have a negative number of half-lives, we take the absolute value:

N ≈ |-0.126| ≈ 0.126

Now we need to calculate the age of the artifact in years. We know that each half-life is 5,730 years. Therefore:

Age ≈ N * Half-life ≈ 0.126 * 5,730

Calculating this, we find that the probable age of the artifact is approximately 721 years.

So, to the nearest 100 years, the probable age of the artifact is 700 years.