The half-life of carbon-14 is 5730 years. How long will it take for 7/8 of a sample of carbon-14 to decay?
11,460 years
17,190 years
22,920 years
28,650 years
The half-life of carbon-14 is 5730 years. To find the time it takes for 7/8 of a sample to decay, we can set up an exponential decay equation.
Let 𝑡 be the time it takes for 7/8 of the sample to decay. The amount of carbon-14 remaining after time 𝑡 is given by:
(1/2)^(𝑡/5730)
Since we want 7/8 of the sample remaining, we can set up the equation:
(1/2)^(𝑡/5730) = 7/8
Taking the natural logarithm (ln) of both sides, we get:
ln((1/2)^(𝑡/5730)) = ln(7/8)
Using the properties of logarithms, we can simplify the equation:
(𝑡/5730) * ln(1/2) = ln(7/8)
Dividing both sides by ln(1/2):
𝑡/5730 = ln(7/8) / ln(1/2)
Simplifying:
𝑡 = (5730 * ln(7/8)) / ln(1/2)
Using a calculator, we find:
𝑡 ≈ 11,460 years
Therefore, it will take approximately 11,460 years for 7/8 of a sample of carbon-14 to decay.
So, the correct answer is 11,460 years.