Question 1195957
.5 = (1+r)^5730
1+r is the annual growth factor.
r is the annual growth rate.
solve for (1+r) to get .5 ^ (1/5730) = 1+r = .9998790392.
confirm this is true because .9998790392 ^ 5730 = .5
if the scroll contains .76 * its original level of carbon-14, then you get:
.76 = .9998790392 ^ x
x is the number of years.
take the log of both sides of the equation to get:
log(.76) = x * log(.9998790392)
solve for x to get:
x = log(.76)/log(.9998790392) = 2268.671315.
that's the number of years to get .76 * the carbon-14 in the artifact to become 76% of what it was when the artifact was new.
that's how old the artifact is presumed to be based on the formula.
this equation can be graphed as shown below.
<img src = "http://theo.x10hosting.com/2022/082201.jpg">
x represents the number of years from when the artifact was created.
y represents the proportion of carbon-14 remaining.
note that .5^(1/5730) is the annual growth factor which we had earlier determined to be .9998790392 rounded to the number of digits that could be displayed by my calculator.