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.

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.

Answer 2

To determine how long ago the person died, we need to solve for x in the equation:

f(x) = A0e^(-0.0001216x)

Given that the amount of carbon-14 present in the mummy is two-fifths (2/5) of the amount found in living human beings, we can set up the following equation:

(2/5)A0 = A0e^(-0.0001216x)

Let's solve for x by isolating it on one side of the equation. First, divide both sides of the equation by A0:

(2/5) = e^(-0.0001216x)

Take the natural logarithm (ln) of both sides to remove the exponential function:

ln(2/5) = ln(e^(-0.0001216x))

Using logarithm properties, we can bring down the exponent:

ln(2/5) = -0.0001216x * ln(e)

Remember that ln(e) is equal to 1, so the equation simplifies to:

ln(2/5) = -0.0001216x

Next, divide both sides of the equation by -0.0001216:

x = (ln(2/5)) / -0.0001216

Using a calculator, we can compute the value of x to find the approximate time since the person died.

Answer 3

To determine how long ago the person died, we need to find the value of x in the exponential function f(x) = A0e^(-0.0001216x), where A0 represents the initial amount of carbon-14 present.

Given that the amount of carbon-14 present in the mummy is about two-fifths (2/5) of the amount found in living human beings, we can set up the equation:

2/5 = e^(-0.0001216x)

To solve for x, we can take the natural logarithm (ln) of both sides of the equation:

ln(2/5) = ln(e^(-0.0001216x))

Using the property of logarithms, ln(e^x) = x, the equation becomes:

ln(2/5) = -0.0001216x

Dividing both sides of the equation by -0.0001216 gives:

x = ln(2/5) / -0.0001216

Using a calculator, we can determine that ln(2/5) is approximately -0.693. Substituting this value into the equation, we get:

x ≈ (-0.693) / -0.0001216

Simplifying, we find:

x ≈ 5692 years

Therefore, the person likely died approximately 5692 years ago.