A bone is found to have 45% of the original amount of carbon 14. If the half life of carbon is 5600 years, how old is the bone.

You could use that formula, if you know how to relate k to half life. In the formula you wrote, there would have to be a negative exponent (assuming k is positive).

Another way that might be easier to remember is to solve:

0.45 = (0.5)^(t/T)
where T is the half life.

Take logs of both sides (to any base)

t/T = log(0.45)/log(0.5) = 1.15

T = 5600/1.15 = 4870 years

yes, you could

You will need
.5 = 1 e^(5600k)
5600k = ln .5
k = ln .5 / 5600

so .45 = e^((ln.5/5600)t
(ln.5/5600)t = ln.45
t = 5600(ln.45)/ln.5
= 6451.2

or the other formula
N = (.5)^(t/5600)

then .45 = .5^(t/5600)
t/5600 = ln.45/ln.5
t = (5600ln.45/ln.5) = same calculation

Thank You

Well, we could use that formula, but let me explain it in a more humorous way.

Imagine the bone as a fossilized comedian who loves carbon dating jokes. So, initially, the bone had a full set of jokes, represented by 100% of the original amount of carbon-14.

But as time passed, the bone's jokes became less funny. So now, it only has 45% of its original comedic ability, or in scientific terms, carbon-14.

Now, the half-life of carbon-14 is 5600 years, which means it takes 5600 years for the bone's jokes (carbon-14) to lose half of their humor.

Using a little math magic, we can figure out how many half-lives the bone has gone through to reach 45%. If each half-life is 5600 years, and the bone has 45% left, then it must have gone through two half-lives, losing half of its humor each time.

So, multiplying 5600 years by 2, we find that the bone is approximately 11,200 years old. That's quite an ancient comedian!

Yes, you can use the formula N = Nₒ * e^(-kt) to solve this problem. In this formula:

- N is the current amount of carbon-14 remaining in the bone,
- Nₒ is the original amount of carbon-14 in the bone,
- k is the decay constant (ln(2) / t₁/₂),
- t is the time (in this case, the age of the bone), and
- t₁/₂ is the half-life of carbon-14.

To find the age of the bone, we need to rearrange the formula and solve for t:

t = (ln(N/Nₒ)) / -k

Given that the bone has 45% of the original amount of carbon-14, we can say:

N = 0.45 * Nₒ

Substituting this value into the formula, we have:

t = (ln(0.45 * Nₒ / Nₒ)) / -k

Simplifying further:

t = (ln(0.45)) / -k

Finally, we need to substitute the value of k, which is given by k = ln(2) / t₁/₂, where t₁/₂ is the half-life of carbon-14 (5600 years in this case):

t = (ln(0.45)) / -(ln(2) / 5600)

Evaluating this expression will give you the age (t) of the bone.