The half-life of carbon-14 is 5600 years. If a piece of charcoal made from the wood of a tree shows only 73% of the carbon-14 expected in living matter, when did the tree die?
Well, it seems like this tree has been out of the dating game for quite some time! According to the half-life of carbon-14, we know that after 5600 years, only 50% of the original amount of carbon-14 remains. So, if the charcoal shows only 73% of the expected amount, we can estimate that it has gone through approximately two half-lives.
To find out when our tree kicked the bucket, we'll need to multiply the half-life (5600 years) by the number of half-lives (2):
5600 years x 2 = 11200 years
So, our hilarious conclusion is that the tree died roughly 11200 years ago. Talk about ancient history!
To determine when the tree died, we can use the concept of half-life and the given information.
The half-life of carbon-14 is given as 5600 years, which means that every 5600 years, the amount of carbon-14 in a sample will be reduced to half.
We are told that the piece of charcoal made from the wood of the tree shows only 73% of the expected carbon-14 in living matter. This means that the amount of carbon-14 has been reduced to 73% of its original quantity.
To find the time when the tree died, we can use the formula:
\[ \text{Final amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{\frac{\text{time}}{\text{half-life}}} \]
Let's denote the time when the tree died as "t". We know that the final amount is 73% (0.73) of the initial amount, and the half-life is 5600 years.
\[ 0.73 = 1 \times \left(\frac{1}{2}\right)^{\frac{t}{5600}} \]
To solve this equation for t, we can take the logarithm base 2 of both sides:
\[ \log_2(0.73) = \frac{t}{5600} \times \log_2(1/2) \]
Using a calculator, we find:
\[ \log_2(0.73) = -0.416 \]
\[ \log_2(1/2) = -1 \]
Substituting these values into the equation:
\[ -0.416 = \frac{t}{5600} \times -1 \]
Simplifying:
\[ t = -0.416 \times 5600 \]
\[ t \approx -2329 \]
The tree died approximately 2329 years before the present time.
To determine when the tree died, we can use the concept of half-life and the information provided.
The half-life of carbon-14 is given as 5600 years, which means that in 5600 years, half of the carbon-14 in a sample will decay.
If the charcoal from the tree shows only 73% of the expected carbon-14, it means that 27% of the carbon-14 has decayed. This 27% represents one half-life of carbon-14.
To find the number of half-lives that have elapsed, we can use the formula:
(Number of Half-lives) = (log(initial amount / final amount)) / (log(0.5))
Let's substitute the values:
(Number of Half-lives) = (log(1 / 0.73)) / (log(0.5))
Calculating this, we find that approximately 0.285 half-lives have elapsed.
Now, we can determine the time by multiplying the number of half-lives by the length of one half-life:
(Time) = (Number of Half-lives) * (Half-life time)
(Time) = 0.285 * 5600 years
Calculating this, we find that the tree died approximately 1606 years ago.
.73 = 1(1/2)^t/5600
.73 = .5^(t/5600)
take log of both sides and use rules of logs
log.73 = (t/5600)log.5
t/5600 = log.73 = log.5 = .45403..
t = appr 2542.6 years