Only 34% of the carbon-14 in a wooden bowl remains. How old is the bowl? (Carbon-14 has a half-life of 5,700 years.)
amount = 1(1/2)^(t/5700)
.34 = .5^(t/5700)
log .34 = (t/5700)log .5
t = 5700(log .34)/log .5 = 8871 years
thank you so much!
To determine the age of the bowl, we can use the concept of half-life and the fact that carbon-14 has a half-life of 5,700 years. The remaining 34% of carbon-14 corresponds to 66% of the original carbon-14.
To calculate the number of half-lives, we can use the formula:
Number of half-lives = (log (percentage remaining) / log (0.5))
Using this formula, we can find the number of half-lives as follows:
Number of half-lives = (log (66%) / log (0.5))
Calculating this:
Number of half-lives ≈ (log(0.66) / log(0.5))
≈ (-0.179 / -0.301)
≈ 0.594
Since each half-life is equivalent to 5,700 years, we can calculate the age of the bowl by multiplying the number of half-lives by the half-life of carbon-14:
Age of the bowl ≈ (0.594 * 5,700)
≈ 3,395.8 years
Therefore, the bowl is approximately 3,395.8 years old.
To determine the age of the wooden bowl, we can use the concept of carbon-14 dating and its half-life. Carbon-14 dating is a method used to estimate the age of organic materials by measuring the amount of carbon-14 remaining in them.
Given that only 34% of the carbon-14 in the wooden bowl remains, we can infer that 66% of the carbon-14 has decayed since the time the bowl was made.
Carbon-14 has a half-life of 5,700 years, which means that every 5,700 years, half of the original carbon-14 atoms decay. Knowing this, we can set up a proportion to determine the age of the bowl:
(66/100) = (1/2)^(n/5700)
Here, 'n' represents the number of half-lives that have occurred since the bowl was made. By solving this equation, we can find the value of 'n', which will indicate the number of half-lives.
Let's solve the equation:
Taking the logarithm of both sides of the equation with base 2 (since we have a fraction with 1/2), we get:
log2(66/100) = (n/5700) * log2(1/2)
Simplifying further, we have:
log2(66/100) = (n/5700) * (-1)
Next, we can isolate 'n' by multiplying both sides by 5700:
n = log2(66/100) * (-5700)
Using a calculator, we can evaluate the right side of the equation to find:
n ≈ -1644.27
Since we cannot have a negative number of half-lives, we take the absolute value of 'n' to get:
|n| ≈ 1644.27
Therefore, approximately 1644 half-lives have occurred since the bowl was made.
To find the age of the bowl, we can multiply the number of half-lives by the length of each half-life, which is 5,700 years:
Age of the bowl ≈ 1644 * 5700 years
Calculating this, we find that the wooden bowl is approximately 9,379,800 years old.