Chapter 7.4, Problem 35E

To determine

To determine the age of Crater Lake.

Answer to Problem 35E

Crater Lake is about $6658.2years$ old or after $6658.2years$ , $44.5\%$ of carbon-14 is left of its original amount.

Explanation of Solution

Given information:

The charcoal from a tree killed in the volcanic eruption that formed Crater lake in Oregon contained $44.5\%$ of the carbon-14 found in living matter.

Calculation:

If $y_0$ is the number of radioactive nuclei present at time zero, the number still present at any later time $t$ will be

  $y=y_0{e}^{-kt}\text{ , }k>0$

Since, it is known that Scientists who use carbon-14 dating use $5700$ years for its half life.

Therefore ,

  $\begin{array}{l}y=y_0{e}^{-kt}\\ \frac{1}{2}{y}_0=y_0{e}^{-5700k}\\ e^{5700k}=2\\ 5700k=\ln 2\\ k=\frac{\ln 2}{5700}\end{array}$

Now let the initial amount of carbon-14 be $100$

  $\begin{array}{l}y=y_0{e}^{-kt}\\ 44.5=100e^{-\frac{\ln 2}{5700}t}\\ t=6658.2years\end{array}$

Therefore, Crater Lake is about $6658.2years$ old or after $6658.2years$ , $44.5\%$ of carbon-14 is left of its original amount.