Question

The amount of​ carbon-14 in an object is given by yequalsa e Superscript negative 0.00012 t​, where a is the amount of​ carbon-14 originally in the​ object, and t is the age of the object in years. A fossil bone contains 24​% of its original​ carbon-14. What is the approximate age of the​ bone?

Answer 1

To find the age of the fossil bone, we start with the formula for the amount of carbon-14 remaining in the object:

\[
y = a e^{-0.00012t}
\]

Here, \(y\) is the remaining amount of carbon-14, \(a\) is the original amount, and \(t\) is the age of the object in years.

In this problem, we know that the fossil bone contains 24% of its original carbon-14. This means:

\[
y = 0.24a
\]

Now we can substitute \(y\) into the equation:

\[
0.24a = a e^{-0.00012t}
\]

We can divide both sides by \(a\) (assuming \(a \neq 0\)):

\[
0.24 = e^{-0.00012t}
\]

Next, we take the natural logarithm of both sides to solve for \(t\):

\[
\ln(0.24) = -0.00012t
\]

Now, solving for \(t\):

\[
t = \frac{\ln(0.24)}{-0.00012}
\]

Calculating \(\ln(0.24)\):

\[
\ln(0.24) \approx -1.4271
\]

Now substitute this value into the equation for \(t\):

\[
t = \frac{-1.4271}{-0.00012} \approx 11892.5
\]

Therefore, the approximate age of the bone is:

\[
t \approx 11893 \text{ years}
\]

So, the age of the fossil bone is approximately 11,893 years.