Only 27% of the carbon-14 in a wooden bowl remains. How old is the bowl? (Carbon-14 has a half-life of 5,700 years.)
Let T be the age in years.
0.27 = (0.5)^(T/5700)
Take logs of both sides to solve.
T/5700 = log(0.27)/log(0.5) = 1.889
T = ___ years
65437990
well the bowel has 27% carbon-14 remaining which means that 73% of the carbon-14 has already been depleted. 5700 years is the half life of carbon-14 so to get the full life we multiply 5700(2)=11400 years. then we can do an easy cross multiplication problem to figure it out, x/11400=73/100 and then we solve for x which is x=11400(73)/100 and the final answer and age of the bowel is 8322 years. a very algebraic way of solving it lol
To determine the age of the wooden bowl, we can use the concept of half-life and the remaining amount of carbon-14.
First, we need to determine how many half-lives have passed. Carbon-14 has a half-life of 5,700 years, which means that after this amount of time, half of the original carbon-14 will have decayed.
Let's denote the original amount of carbon-14 in the bowl as "B0". Since only 27% of carbon-14 remains, we know that 27% of the original amount is equal to the remaining amount, which we can denote as "Bt".
So, Bt = 0.27 * B0
Since each half-life reduces the amount of carbon-14 by half, the number of half-lives, denoted as "n," can be calculated using the following equation:
0.5^n * B0 = Bt
Let's substitute the value of Bt we found earlier:
0.5^n * B0 = 0.27 * B0
Next, divide both sides of the equation by B0:
0.5^n = 0.27
Now, we can solve for "n" by taking the logarithm of both sides with base 0.5:
log0.5 (0.5^n) = log0.5 (0.27)
Since log0.5 (0.5^n) is equal to "n," we get:
n ≈ log0.5 (0.27)
Using a calculator, we find that n ≈ 1.59.
Since "n" represents the number of half-lives, we can round it to the nearest whole number, which is 2.
Now, to determine the age of the bowl, we multiply the number of half-lives by the half-life period of carbon-14:
Age = n * Half-life
Age ≈ 2 * 5,700 years
The wooden bowl is approximately 11,400 years old.