Carbon-14 Dating: When Did the Iceman Die?

[Meeeessttteeehh]

Homework Statement

Carbon-14 testing on the body of the Iceman showed that the level of carbon-14 was at 52%. In approximately what year did he die?

Homework Equations

We have been given no equations, and the course work is vague. We have to make charts... (grade 11)

The Attempt at a Solution

I know that carbon-14 has a half-life of 5730 years (which is confusing because my course switches between 6000 and 5730 years...), but I am not sure how to get 52%, and to be completely honest, I am lost when it comes to calculating dates before year 1... If I round to 50% that means he died about 5730 years ago, and our lesson says he was discovered in 1991. I am assuming this carbon dating was done in 1991, so 1991 - 5730 years? That being said, I don't think that 50% is quite specific enough... if I had the "whole life" of carbon-14 I could multiply by 0.52 and that would be the percentage, but since each half-life just divides in half, I don't see it possible for there to ever be zero. (This is like those curves in math that never cross the x-axis?!) Not sure where to go from here, and any help is appreciated!

 

[cnh1995]

(This is like those curves in math that never cross the x-axis?!)

Right, it's an exponential decay.
You need to write the differential equation that describes this situation. You can use the known half-life to get its exact solution.

 

[Meeeessttteeehh]

I'm not sure what you mean by differential equation... we weren't taught any equations...
I talked to my teacher and he said something confusing about the 52% and such. Basically I understood that I cannot multiply the 5730 by 0.52 because that's only half of the half... If I multiply it by 0.96 it should work (not positive why, especially since that's less than all of the half and I need more than half... oh my). Then I just subtract the number from 1991. So I have 1991 - 5500.8 which is about 3510 BC? If I used 96% of the rounded half-life of carbon-14 i get 6000 x 0.96 = 5760 (I think that's it!) so 1991 - 5760 = 3769 BC. Do either of these make sense? SOS Little Sheldon! XD

 

[cnh1995]

I'm not sure what you mean by differential equation... we weren't taught any equations...
I talked to my teacher and he said something confusing about the 52% and such. Basically I understood that I cannot multiply the 5730 by 0.52 because that's only half of the half... If I multiply it by 0.96 it should work (not positive why, especially since that's less than all of the half and I need more than half... oh my). Then I just subtract the number from 1991. So I have 1991 - 5500.8 which is about 3510 BC? If I used 96% of the rounded half-life of carbon-14 i get 6000 x 0.96 = 5760 (I think that's it!) so 1991 - 5760 = 3769 BC. Do either of these make sense? SOS Little Sheldon! XD

Well, the differential equation will give you the exact year of the iceman's death, but since you haven't studied it yet, let's drop this approach.

In approximately what year did he die?

You are supposed to find an approximate answer and 52% is very close to 50%.

 

[jbriggs444]

I know that carbon-14 has a half-life of 5730 years (which is confusing because my course switches between 6000 and 5730 years...), but I am not sure how to get 52%

If 50% versus 52% is not close enough, one can attack this problem as follows.

The fraction of carbon (f) remaining after 5730 years is 1/2. It's an exponential decay. So the formula for fraction remaining is: $f = 2^{-\frac{years}{5730}}$

You are given that f = 52% and you want to solve for years. Start by taking the base 2 log of both sides.

Tip: $log_2\ x = \frac{log\ x}{log\ 2}$

 

[Meeeessttteeehh]

Okay, thanks for the input guys! I wrote this:

Sound reasonable?

 

[haruspex]

Okay, thanks for the input guys! I wrote this:
View attachment 219390
Sound reasonable?

Maybe, but I feel that since they specified 52% you should attempt to use that.
Note that it is not exactly proportional. 52 v. 50 is a 4% difference, but it makes a greater percentage difference to the age.
Do try to use the tips you have already been given.

 

[Humbleness]

My attempt at the question...
If 50% = 5730 years
1% = 114.6 years
2% = 229.2 or approximately 230 years
5730 - 230 = 5500 years
Therefore, 5500 years before 1991 (The year Iceman was discovered), meaning 5500-1991 = 3509 years or 3510 BC approximately
Would this be good?

 

[Orodruin]

My attempt at the question...
If 50% = 5730 years
1% = 114.6 years
2% = 229.2 or approximately 230 years
5730 - 230 = 5500 years
Therefore, 5500 years before 1991 (The year Iceman was discovered), meaning 5500-1991 = 3509 years or 3510 BC approximately
Would this be good?

No. You seem to be assuming a linear decay, which is simply wrong. The decay is exponential. Also, providing solutions before a problem has been explicitly solved by the OP is against forum rules.