SOLUTION: Hi please assist me with this question Radioactive carbon-14 is used to determine the age of artifacts because it concentrates in the organism only when they are alive. It has

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Question 1195957: Hi please assist me with this question
Radioactive carbon-14 is used to determine the age of artifacts because it concentrates in the
organism only when they are alive. It has a half-life of 5730 years. In 1947, Dead Sea Scroll
were found by one of the South Africa Universities professor. Analysis indicate that scroll
wrapping contain s 76% of their original carbon -14. Estimate the age of the Dead Sea Scroll.
[10 marks]
Found 4 solutions by Theo, greenestamps, josgarithmetic, MathTherapy:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
.5 = (1+r)^5730
1+r is the annual growth factor.
r is the annual growth rate.
solve for (1+r) to get .5 ^ (1/5730) = 1+r = .9998790392.
confirm this is true because .9998790392 ^ 5730 = .5
if the scroll contains .76 * its original level of carbon-14, then you get:
.76 = .9998790392 ^ x
x is the number of years.
take the log of both sides of the equation to get:
log(.76) = x * log(.9998790392)
solve for x to get:
x = log(.76)/log(.9998790392) = 2268.671315.
that's the number of years to get .76 * the carbon-14 in the artifact to become 76% of what it was when the artifact was new.
that's how old the artifact is presumed to be based on the formula.
this equation can be graphed as shown below.

x represents the number of years from when the artifact was created.
y represents the proportion of carbon-14 remaining.
note that .5^(1/5730) is the annual growth factor which we had earlier determined to be .9998790392 rounded to the number of digits that could be displayed by my calculator.

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The fraction of carbon-14 remaining after n half-lives is %281%2F2%29%5En. Use that formula to determine the number half-lives.

%280.5%29%5En=0.76

The variable is in an exponent, so use logarithms.

log%28%280.5%29%5En%29=log%280.76%29
n%2Alog%280.5%29=log%280.76%29
n=log%280.76%29%2Flog%280.5%29 = 0.396 to 3 decimal places

Multiply the number of half-lives by the number of years in a half-life.

0.396%285740%29=2269

ANSWER: The age of the Dead Sea Scrolls is about 2269 years.

Note that, as the problem says, this is an ESTIMATE of the age of the scrolls.

Radioactive decay is a statistical process; the rate of decay is not absolutely constant. So any calculation of the age of an object using carbon-14 dating only gives an approximate answer. So this is an example of a calculation where you do NOT want to keep a large number of decimal places in your calculations.

I found one internet source that says if the age is between 1000 and 10,000 years the convention is to round the age to the nearest 10. So the best answer to this problem is that the age is ABOUT 2270 years.

Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Half-Life, 5730 years

y=pe%5E%28-kx%29
-
ln%28y%29=ln%28p%29-kx%2Aln%28e%29
ln%28y%29=ln%28p%29-kx
-kx=ln%28y%29-ln%28p%29
k=%28ln%28p%29-ln%28y%29%29%2Fx
from HALF-LIFE information, k=ln%282%29%2F5730
highlight_green%28k=0.000120968%29

The form of the model you want to use is from x=%28ln%28y%29-ln%28p%29%29%2F%28-k%29
x=%281%2Fk%29%28ln%28p%29-ln%28y%29%29
highlight_green%28x=%281%2Fk%29%28ln%28p%2Fy%29%29%29

Using the given values, x=%28ln%28100%2F76%29%29%2F0.000120968
highlight%28x=2270%29

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
Hi please assist me with this question
Radioactive carbon-14 is used to determine the age of artifacts because it concentrates in the
organism only when they are alive. It has a half-life of 5730 years. In 1947, Dead Sea Scroll
were found by one of the South Africa Universities professor. Analysis indicate that scroll
wrapping contain s 76% of their original carbon -14. Estimate the age of the Dead Sea Scroll.
[10 marks]
If ½ life is “a” time-periods, then k, or DECAY CONSTANT = matrix%281%2C3%2C+ln%281%2F2%29%2Fa%2C+or%2C+ln%28.5%29%2Fa%29 

                                          We then get: 

                     Continuous Growth/Decay formula: 
                                                     matrix%281%2C3%2C+.76%2C+%22=%22%2C+e%5E%28-.000121t%29%29 ------ Substituting .76 for A%2FA%5Bo%5D, and -.000121 for k
                                              matrix%281%2C3%2C+-.000121t%2C+%22=%22%2C+ln+%28.76%29%29 ------ Converting to NATURAL LOGARITHMIC (ln) form 
            
              Estimated age of the Dead Sea Scroll, or