Chapter 3.5, Problem 37E

a.

To determine

Find the age of the piece of wood.

Answer to Problem 37E

The age of the piece of wood is $12170.04$ years.

Explanation of Solution

Given:

The ratio of $C_{14}$ to $C_{12}$ in a piece of wood discovered in a cave is $R=\frac{1}{8^{14}}$ .

Calculation:

Use carbon dating model.

  $R=\frac{1}{{10}^{12}}{e}^{-\frac{t}{8223}}$

Substitute the value of $R=\frac{1}{8^{14}}$ in the equation.

  $\begin{array}{l}\Rightarrow \frac{1}{8^{14}}=\frac{1}{{10}^{12}}{e}^{-\frac{t}{8223}}\\ \Rightarrow \frac{1}{8^{14}}\cdot {10}^{12}=\frac{1}{{10}^{12}}\cdot {10}^{12}{e}^{-\frac{t}{8223}}\\ \Rightarrow \frac{{10}^{12}}{8^{14}}=e^{-\frac{t}{8223}}\\ \Rightarrow 0.227=e^{-\frac{t}{8223}}\end{array}$

Take natural log on both sides.

  $\begin{array}{l}\Rightarrow \ln 0.227=\ln e^{-\frac{t}{8223}}\\ \Rightarrow -1.48=-\frac{t}{8223}\\ \Rightarrow 1.48=\frac{t}{8223}\\ \Rightarrow 1.48\cdot 8223=\frac{t}{8223}\cdot 8223\\ \Rightarrow 12170.04\approx t\\ \Rightarrow t\approx 12170.04\end{array}$

Hence the age of the piece of wood is $12170.04$ years.

b.

To determine

Find the age of the piece of paper.

Answer to Problem 37E

The age of the piece of paper is $4797.48$ years.

Explanation of Solution

Given:

The ratio of $C_{14}$ to $C_{12}$ in a piece of paperburied in a tomb is $R=\frac{1}{{13}^{11}}$ .

.Calculation:

Use carbon dating model.

  $R=\frac{1}{{10}^{12}}{e}^{-\frac{t}{8223}}$

Substitute the value of $R=\frac{1}{{13}^{11}}$ in the equation.

  $\begin{array}{l}\Rightarrow \frac{1}{{13}^{11}}=\frac{1}{{10}^{12}}{e}^{-\frac{t}{8223}}\\ \Rightarrow \frac{1}{{13}^{11}}\cdot {10}^{12}=\frac{1}{{10}^{12}}\cdot {10}^{12}{e}^{-\frac{t}{8223}}\\ \Rightarrow \frac{{10}^{12}}{{13}^{11}}=e^{-\frac{t}{8223}}\end{array}$

Take natural log on both sides.

  $\begin{array}{l}\Rightarrow \ln \frac{{10}^{12}}{{13}^{11}}=\ln e^{-\frac{t}{8223}}\\ \Rightarrow -0.5834218=-\frac{t}{8223}\\ \Rightarrow 0.5834218=\frac{t}{8223}\\ \Rightarrow 0.5834218\cdot 8223=\frac{t}{8223}\cdot 8223\\ \Rightarrow 4797.48\approx t\\ \Rightarrow t\approx 4797.48\end{array}$

Hence the age of the piece of paper is $4797.48$ years.