Question

Scientist can determine the age of ancient objects by a method called radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, 14C, with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates 14C through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of 14C begins to decrease through radioactive decay. Therefore, the level of radioactivity must also decay exponentially. A parchment fragment was discovered that had about 64% as much 14C radioactivity as does plant material on Earth today. Estimate the age of the parchment. (Round your answer to the nearest hundred years.) yr

Answer 1

Answer:

$3688.323$years

Explanation:

Given-

Half life of $14$C $= 5730$years

As we know -

$A_{(t)} = A_0e^{kt}$

Where

$A_{(t)} =$ Mass of radioactive carbon after a time period "t"

$A_0=$ initial mass of radioactive carbon

$k =$radioactive decay constant

$t =$time

First we will find the value of "k"

$\frac{1}{2} = (1)*e^{k*5730}$

On solving, we get -

$e^{5730*k}= 0.5\\5730*k = ln(0.5)\\k = -0.000121$

Now, when mass of 14C becomes $64$% of the plant  material on earth today, then its age would be

$A_{(t)} = A_0*e^{(-0.000121*t)}\\A_{(t)}= 0.64*A_0\\0.64*A_0 = A_0*e-^{(0.000121*t)}\\t = 3688.323$years