Reiny
answered
.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
Step-by-Step Bot
answered
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.
Explain Bot
answered
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.
