The half life of Carbon-14 is 5730 years.
a. find the rule of the function that gives the amount remaining from an initial quantity of 100 milligrams of carbon14 after t years.
b. the burial cloth of an egyptian mummy i sestimated to contain 59% of the carbon 14 it contained originally. how long ago was the mummy buried?
amount = 100 (1/2)^(t/5730)
b)
59 = 100 (.5)^(t/5730)
.59 = .5^(t/5730)
ln .59 = (t/5730) ln (.5)
t/5730 = ln .59/ln .5 = .76121314
t = 4361.75 yrs
a. To find the rule of the function that gives the amount remaining from an initial quantity of 100 milligrams of carbon-14 after t years, we need to consider the concept of exponential decay.
The formula for exponential decay is given by:
A = A0 * (1/2)^(t / hl)
where:
- A is the amount remaining after t years,
- A0 is the initial quantity (100 milligrams in this case),
- t is the time in years, and
- hl is the half-life of Carbon-14 (5730 years).
Using this formula, we can substitute the given values to find the rule of the function:
A = 100 * (1/2)^(t / 5730)
b. To calculate how long ago the Egyptian mummy was buried, we can use the fact that the burial cloth contains 59% of the original Carbon-14.
Let's denote the time elapsed since the mummy was buried as T years. According to the formula, after T years, the amount remaining should be 59% of the initial quantity (100 milligrams):
0.59 = 100 * (1/2)^(T / 5730)
To solve for T, we need to isolate the exponential term:
(1/2)^(T / 5730) = 0.59 / 100
Next, we can take the logarithm of both sides to get rid of the exponent:
log((1/2)^(T / 5730)) = log(0.59 / 100)
T / 5730 * log(1/2) = log(0.59 / 100)
Finally, we solve for T by multiplying both sides by 5730 and dividing by log(1/2):
T = (log(0.59 / 100) * 5730) / log(1/2)
By evaluating this expression using a calculator, we can determine how long ago the mummy was buried.