The burial cloth of an Egyptian mummy is estimated to contain 57% of the carbon-14 it contained originally. How long ago was the mummy buried? (The half-life of carbon-14 is 5730 years.)
solve
.57 = 1(1/2)^(t/5730)
log both sides
log .57 = log (.5)^(t/5730)
log .57 = t/5730 (log .5)
t/5730 = log .57/log .5
t/5730 = .810966
t = 4646.8
appr. 4647 years
To determine how long ago the mummy was buried, we can use the concept of half-life and the given information.
Carbon-14 has a half-life of 5730 years, which means that after 5730 years, half of the Carbon-14 in a sample will have decayed.
Let's assume that the original amount of Carbon-14 in the burial cloth was 100%. According to the information given, it now contains 57% of its original amount.
Using the half-life concept, we can calculate the number of half-lives that have passed:
(100% -> 50% -> 25% -> 12.5% -> ... -> 57%)
To go from 100% to 57%, we need to calculate the number of half-lives it takes.
57% is closer to 50% than to 100%, so we know that less than one half-life has passed.
To find out the exact number of half-lives, we can use a logarithmic equation:
57% = 50% x (1/2)^n,
Where n represents the number of half-lives.
Dividing both sides by 50% gives:
57% / 50% = (1/2)^n.
Simplifying, we have:
1.14 = (1/2)^n.
To isolate n, we can take the logarithm (base 2) of both sides:
log2(1.14) = log2((1/2)^n).
Using logarithmic rules, we can bring the exponent down:
log2(1.14) = n x log2(1/2).
Given that log2(1/2) = -1, we can substitute it into the equation:
log2(1.14) = n x -1.
Finally, we solve for n:
n = log2(1.14) / -1.
Using a calculator, we find:
n ≈ -0.1547.
Since n represents the number of half-lives, and we can't have a negative number of half-lives, we take the absolute value:
|n| ≈ 0.1547.
Now we multiply the absolute value of n by the half-life of Carbon-14:
Time = |n| x half-life.
Time ≈ 0.1547 x 5730 years.
Calculating, we find:
Time ≈ 885.981 years.
Therefore, the mummy was buried approximately 886 years ago.
To determine how long ago the mummy was buried, we can use the concept of radioactive decay and the half-life of carbon-14.
The half-life of carbon-14 is the time it takes for half of the radioactive carbon-14 atoms in a sample to decay. In this case, it is 5730 years.
The formula to calculate the age of a sample using its remaining percentage of carbon-14 is:
age = (half-life) x log(base 1/2) (remaining percentage / starting percentage)
In this case, the starting percentage is 100%, and the remaining percentage is 57%.
Using the above information, we can substitute the values into the formula:
age = (5730 years) x log(base 1/2) (0.57 / 1)
To calculate the logarithm, we need to convert the base to a common logarithm (base 10). The formula becomes:
age = (5730 years) x log(base 10) (0.57) / log(base 10) (1/2)
Using a scientific calculator or math software, calculate log(base 10) (0.57). The result is approximately -0.243.
Now, calculate log(base 10) (1/2). The result is approximately -0.301.
Substitute these values back into the age formula:
age ≈ (5730 years) x (-0.243) / (-0.301)
Simplifying further:
age ≈ 4648 years
Therefore, the mummy was buried approximately 4648 years ago.