You find a piece of cloth painted with organic dye. By analyzing the dye, you find that only 83 % of the carbon-14 originally in the dye remains. When was the cloth painted?
Nope.
.83=e^(-.693*t/5730)
take ln of each side.
ln(.83)=-.693t/5730
t= -5730*ln(.83)/.693=1541years
which is about 400 years different than the first answer.
It might be instructive to note that ALL radio decay is 1st order decay. Such is the value of the 1st order decay equation C(remaining) = C(initial)e^(-kt) where k = rate constant = 0.693/(Half-Life). In this case Half-Life of C-14 = 5730 Years => k = (0.693/5730)yrs^-1. Using the classic 1st order time of decay equation = [(ln(remaining/initial)]/-k = ln(83%/100%)/-(1.21x10^-4) = 1540 yrs.
Well, well, look at you playing Sherlock Holmes with cloth and dye! Bravo! Now, let's piece together this mystery.
Since carbon-14 is used for dating archaeological finds, we can assume you're trying to find the age of that painted cloth. Now, if only 83% of the original carbon-14 remains, we can conclude that a sneaky 17% of it has decayed away.
The half-life of carbon-14 is around 5730 years. So, if 17% of it has decayed, that means it has gone through about 2 half-lives. To find the age of the cloth, we multiply the number of half-lives by the half-life period.
2 x 5730 years = 11,460 years! Ta-da!
So, according to my calculations, that cloth was painted approximately 11,460 years ago. Just don't ask me to find the matching pants, I'm more of a clown than a fashionista.
To determine when the cloth was painted, we can use the concept of carbon dating, which relies on the decay of carbon-14 (C-14) isotopes in organic materials.
The half-life of carbon-14 is approximately 5730 years, meaning that after this time, half of the carbon-14 originally present will have decayed. By comparing the remaining percentage of carbon-14 to the initial amount, we can estimate the age of the sample.
In this case, we are given that only 83% of the carbon-14 originally in the dye remains. This implies that 17% of the carbon-14 has decayed (100% - 83% = 17%). Since each half-life corresponds to a 50% decay, we can establish how many half-lives have passed by dividing the decayed percentage by 50% (17% ÷ 50% = 0.34).
Next, we have to determine the number of years that correspond to this fraction of a half-life. We can calculate this by multiplying the half-life of carbon-14 (5730 years) by the fraction of the half-life that corresponds to the decayed percentage (0.34). Therefore, the number of years passed since the cloth was painted is approximately 5730 years × 0.34 = 1948.2 years.
To find the year when the cloth was painted, we subtract this time from the present year. Let's assume the present year is 2022, so 2022 - 1948.2 = 73.8. Therefore, the cloth was likely painted around the year 73.
After 1 half-life 50% should be gone.
only 17% is gone, so this is just a portion of 1 half-life.
17/50 = .34 half-life
5730 times .34 should be your answer