the amount of carbon-14 in a mammoth's bone is 45% of the amount found in a living organism. How long ago did the mammoth die? the half-life of carbon-14 is about 5,730 years.
.5 = e^-k(5730)
ln .5 = -.6931 = - 5730 k
k = 1.2097*10^-4
.45 = e^-1.2097*10^-4 t
ln .45 = -.7985 = -1.2097*10^-4 t
t = 6601 years
or, you can think of it like this: every half-life, 1/2 of what was there is gone. So, the amount left after t half-lives is
(1/2)^(t/5730)
So, we want t when
(1/2)^(t/5730) = .45
t/5730 = log(.45)/log(.5) = 1.152
That is, it takes 1.152 half-lives to reduce to 45%
So, t = 1.152 + 5730 = 6601
To determine how long ago a mammoth died based on the amount of carbon-14 remaining in its bone, we can use the concept of half-life. The half-life of carbon-14 is about 5,730 years, which means that every 5,730 years, the amount of carbon-14 in a sample is reduced by half.
Given that the amount of carbon-14 in the mammoth's bone is 45% of the amount found in a living organism, we can set up the following equation:
Amount of carbon-14 remaining = Initial amount of carbon-14 * (0.45) ^ (number of half-lives)
We want to solve for the number of half-lives, which represents the time that has elapsed since the mammoth died. Let's denote this value as "x."
0.45 = 1 * (0.5) ^ x
To solve for x, we can take the logarithm of both sides of the equation:
log(0.45) = x * log(0.5)
Using base 10 logarithm, we can solve for x:
x = log(0.45) / log(0.5)
x ≈ 0.2061 / (-0.3010)
x ≈ -0.685
The result is approximately -0.685. However, it doesn't make sense to have a negative number of half-lives, so we can disregard the negative sign.
Therefore, the mammoth died approximately 0.685 half-lives ago. To find the corresponding time, we need to multiply this value by the half-life of carbon-14:
Time since mammoth died ≈ 0.685 * 5,730 years
Time since mammoth died ≈ 3,925 years
Therefore, the mammoth died approximately 3,925 years ago.