Question

Carbon-14 is a radioactive form of carbon that is found in all living plants and animals. After a plant or animal​ dies, the radiocarbon disintegrates. Scientists determine the age of the remains by comparing the amount of​ carbon-14 present with the amount found in living plants and animals. The amount of​ carbon-14 present after x years is modeled by the exponential function f(x)=A0e−0.0001216x​, where A0 represents the initial amount.
Suppose a mummy from a certain country is discovered in which the amount of​ carbon-14 present is only about two​-fifths the amount found in living human beings. About how long ago did the person​ die?

Answer 1

To find out how long ago the person died, we need to find the value of x when f(x) is equal to two-fifths (2/5) of the initial amount, A0.

We can set up the equation:
2/5 = A0 * e^(-0.0001216x)

Let's solve for x:
2/5 = e^(-0.0001216x)

To solve for x, we can take the natural logarithm of both sides of the equation:
ln(2/5) = ln(e^(-0.0001216x))

Using the properties of logarithms, we can bring down the exponent:
ln(2/5) = -0.0001216x

Now, we can isolate x:
x = (ln(2/5))/(-0.0001216)

Using a calculator, we can evaluate the right side to find:
x ≈ 11476.7 years

Therefore, the person died approximately 11,476.7 years ago.