A wooden artifact from an ancient tomb contains 55% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years. Round your answer to the nearest whole number.)
1((1/2)^(t/5730) = .55
(.5)^(t/5730 = .55
take log of both sides
log (.5)^(t/5730 = log .55
(t/5730) log .5 = log .55
t/5730 = log .55/log .5
t = 5730(log .55/log .5)
= appr 4942 years
Well, let me calculate that using my hilarious math skills!
If the artifact only has 55% of the carbon-14 compared to living trees, I guess you could say it's feeling a bit "half-lifeless."
Now, since the half-life of carbon-14 is 5730 years, we can imagine the artifact saying, "I'm only halfway there, babe." So, if it's halfway there, it must have been made about 5730 years ago!
Now, I know rounding can be a little tricky, but let's just pretend that Mr. Artifact is a round character and round our answer to the nearest whole number. Voila! The artifact was made approximately 5730 years ago.
Oh, the things we can learn from ancient wooden objects and a sprinkle of humor!
To determine how long ago the wooden artifact was made, we can use the concept of half-life.
The half-life of carbon-14 is 5730 years, which means that after 5730 years, half of the carbon-14 in a sample will decay.
Given that the wooden artifact contains 55% of the carbon-14 found in living trees, we can set up the following equation:
0.55 = (1/2)^(t/5730)
Where 't' represents the time in years since the artifact was made.
To solve for 't', take the logarithm of both sides of the equation:
log(0.55) = (t/5730) * log(1/2)
Simplifying the equation, we have:
t/5730 = log(0.55) / log(1/2)
Now, we can solve for 't':
t = (log(0.55) / log(1/2)) * 5730
Using a calculator, we find:
t ≈ 2139
Therefore, the artifact was made approximately 2139 years ago.
To determine how long ago the artifact was made, we need to use the concept of carbon-14 dating.
Carbon-14 is an isotope of carbon that is present in the atmosphere and absorbed by living organisms through respiration and photosynthesis. When an organism dies, it stops taking in carbon-14, and the amount of carbon-14 begins to decrease over time due to radioactive decay.
The half-life of carbon-14 is 5730 years, which means that every 5730 years, half of the amount of carbon-14 in a sample will decay. In other words, after one half-life, 50% of the carbon-14 remains, after two half-lives, 25% remains, and so on.
In this case, the wooden artifact contains 55% of the carbon-14 that is present in living trees. This means that the remaining 45% has decayed since the artifact was made.
To find out how many half-lives have passed, we can use the formula:
number of half-lives = (ln(percentage remaining) / ln(0.5))
Plugging in the values:
number of half-lives = (ln(0.45) / ln(0.5))
Using a scientific calculator:
number of half-lives ≈ 0.5596 / 0.6931 ≈ 0.805
Since we round to the nearest whole number, this tells us that approximately 1 half-life has passed since the artifact was made.
Now, we need to determine the time it takes for one half-life to pass, which is 5730 years according to the information given. Therefore, to find the age of the artifact, we need to multiply the number of half-lives by the length of one half-life.
Age of the artifact = number of half-lives * length of one half-life
Age of the artifact ≈ 1 * 5730 ≈ 5730 years
Therefore, the artifact was made approximately 5730 years ago.