An artifact was found and tested for its carbon-14 content. If 77% of the original carbon-14 was still present, what is its probable age (to the nearest 100 years)? Use that carbon-14 has a half-life of 5,730 years.
Since this is a half-life type, we can use the base of 1/2
.77 = (1/2)^(t/5730)
ln .77 = ln (.5^(t/5730))
t/5730 = ln.77/ln .5
t = 5730(ln.77/ln.5) = 2160.6
to the nearest 100 years that would be 2200 years.
The atmospheric pressure P in pounds per square inch (psi) is given by the formula below, where a is the altitude above sea level (in miles). If a city has an atmospheric pressure of 13.25 psi, what is its altitude? (Recall that 1 mi = 5,280 ft. Round your answer to the nearest foot.)
To calculate the probable age of the artifact, we can use the concept of the half-life of carbon-14. The half-life of carbon-14 is 5,730 years, which means that after 5,730 years, half of the carbon-14 in a sample will have decayed.
Since 77% of the original carbon-14 content is still present, we can determine the number of half-lives that have passed. Let's assume the original amount of carbon-14 was 100 units. If 77% is still present, then there are 77 units of carbon-14 remaining.
To find the number of half-lives, we can set up the equation:
77 = 100 * (0.5)^(n)
Where 'n' is the number of half-lives. Rearranging the equation gives us:
0.77 = 0.5^n
To solve for 'n', we can take the logarithm (base 0.5) of both sides:
log(0.77) = n * log(0.5)
Using a calculator, we find that log(0.77) ≈ -0.1133 and log(0.5) = -0.3010. Plugging these values into the equation, we have:
-0.1133 = n * (-0.3010)
Simplifying further, we get:
n ≈ -0.1133 / -0.3010 ≈ 0.376
Since 'n' represents the number of half-lives, we can round up to the nearest whole number since there can't be a fraction of a half-life. Therefore, 'n' is approximately 1.
Now that we know 1 half-life has passed, we can multiply it by the half-life of carbon-14 (5,730 years) to find the probable age:
Age ≈ 1 * 5,730 years ≈ 5,730 years
Hence, the probable age of the artifact is approximately 5,730 years.