Carbon-14 is used for archaeological carbon dating. Its half-life is 5,730 years. How much of a 100g sample of carbon-14 will be left in 1,000 years? Write your answer rounded to nearest hundredth. A≈88.61g

This problem requires two main steps. First, we must find the decay constant k. If we start with 100g, at the half-life there will be 50g remaining. We will use this information to find k. Then, use that value of k to help us find the amount of sample that will be left in 1,000 years.
Identify the variables in the formula.
AA0ktA=50=100=?=5,730years=A0ekt
Substitute the values in the formula.
50=100ek⋅5,730
Solve for k. Divide each side by 100.
0.5=e5,730k
Take the natural log of each side.
ln0.5=lne5,730k
Use the power property.
ln0.5=5,730klne
Simplify.
ln0.5=5,730k
Divide each side by 5,730.

ln0.55,730=k (exact answer)

We use this rate of growth to predict the amount that will be left in 1,000 years.
AA0ktA=?=100=ln0.55,730=1,000 years=A0ekt
Substitute in the values.
A=100eln0.55,730⋅1,000
Evaluate.

A≈88.61g

In 1,000 years, there will be approximately 88.61g remaining.