a fragment of a bone is discovered to 20% of the usual C-14 concentration.estimate the age of the bone,given that half life time of C-14 is5730years
The half life tells us that after 5730 years, half the original substance remains:
( 1 / 2 ) a = a * e ^ ( r * 5730 ) Divide both sides by a
1 / 2 = e ^ ( r * 5730 ) Take the natural logarithm of both sides
ln ( 1 / 2 ) = r * 5730 Divide both sides by 5730
ln ( 1 / 2 ) / 5730 = r
r = ln ( 1 / 2 ) / 5730
ln ( 1 / 2 ) = - ln 2
so:
r = - ln ( 2 ) / 5730
r = - 0.6931471806 / 5730
r = - 0.0001201
The decay will follow the equation:
Q(t) = a * e ^ (− 0.000121 t )
To find how old the bone fragment is that contains 20% of the original amount, we solve for t when Q(t) = 0.20 a
0.2 a = a * e ^ (− 0.000121 t ) Divide both sides by a
0.2 = e ^ (− 0.000121 t ) Take the natural logarithm of both sides
ln ( 0.2 ) = − 0.000121 t Divide both sides by − 0.000121
ln ( 0.2 ) / − 0.000121 = t
t = ln ( 0.2 ) / − 0.000121
t = -1.6094379124341 / − 0.000121
t = 13301.1397722 yrs
approx. 13 300 yrs
Well, I'm not an archaeologist, but I'll give it a shot. Let's see, if the fragment of the bone has 20% of the usual C-14 concentration, we can assume that it has undergone about 80% of its decay. Since the half-life of C-14 is 5730 years, we can estimate that it has gone through about 3 half-lives. So, we can multiply the half-life by the number of half-lives to get an estimate: 5730 years x 3 = 17,190 years! But take my estimate with a pinch of clownish skepticism. Remember, the actual age may vary!
To estimate the age of the bone fragment, we can use the concept of exponential decay and the half-life of carbon-14.
The half-life of carbon-14 is 5730 years, which means that after every 5730 years, the amount of carbon-14 in a sample is reduced by half.
In this case, if the bone fragment is discovered to have 20% of the usual C-14 concentration, it means that the remaining concentration of carbon-14 is 20% of the initial amount.
Let's assume that the initial concentration of carbon-14 in the bone fragment was 100 units (just for easy calculations). If we have 20% remaining, it means we currently have 20 units of carbon-14.
Now, using the half-life of carbon-14, we can calculate the number of half-lives that have passed since the bone was alive:
100 units ➡️ 1 half-life ➡️ 50 units
50 units ➡️ 1 half-life ➡️ 25 units
25 units ➡️ 1 half-life ➡️ 12.5 units
12.5 units ➡️ 1 half-life ➡️ 6.25 units
6.25 units ➡️ 1 half-life ➡️ 3.125 units
3.125 units ➡️ 1 half-life ➡️ 1.5625 units
1.5625 units ➡️ 1 half-life ➡️ 0.78125 units
0.78125 units ➡️ 1 half-life ➡️ 0.390625 units
0.390625 units ➡️ 1 half-life ➡️ 0.1953125 units
0.1953125 units ➡️ 1 half-life ➡️ 0.09765625 units
0.09765625 units ➡️ 1 half-life ➡️ 0.048828125 units
0.048828125 units ➡️ 1 half-life ➡️ 0.0244140625 units
At this point, we can see that the remaining concentration of carbon-14 (0.0244140625 units) is less than 0.5 units, which is the limit we are considering for our calculations. Therefore, the number of half-lives that have passed would be 11.
Since the half-life of carbon-14 is 5730 years, we can calculate the age of the bone fragment by multiplying the number of half-lives by the half-life:
11 half-lives * 5730 years/half-life = 63,030 years
Therefore, the estimated age of the bone fragment is approximately 63,030 years.
To estimate the age of the bone, we can use the concept of half-life of Carbon-14 (C-14).
The half-life of C-14 is 5730 years, which means that after 5730 years, half of the C-14 in a sample will have decayed.
In this case, the fragment of the bone is discovered to have 20% of the usual C-14 concentration.
Let's assume that the original C-14 concentration in the bone fragment was "x".
After one half-life (5730 years), the remaining C-14 concentration becomes half of "x", i.e., 0.5x.
After two half-lives (2 * 5730 years), the remaining C-14 concentration becomes half of 0.5x, i.e., 0.25x.
Given that the fragment has 20% of the usual C-14 concentration, we can set up the following equation:
0.2x = 0.25x
Simplifying the equation:
0.2x - 0.25x = 0
-0.05x = 0
x = 0 / -0.05
x = 0
From the equation, we find that the original C-14 concentration in the bone fragment was 0.
This suggests that either the bone fragment is contaminated or the C-14 dating method cannot be applied to this particular bone fragment.