Chapter 30, Problem 59GP

To determine

The age of the tool.

Answer to Problem 59GP

The age of the tool is $23263.8\text{years}$ .

Explanation of Solution

Given:

The old wooden tool contains $6\%$ of ${}_6^{14}\text{C}$ in comparison of the fresh wood. The half-life of carbon-14 is 5730 years.

Formula used:

Write the expression to calculate the number of half-lives.

  $\frac{N}{N_o}={\left(\frac{1}{2}\right)}^n$.... (1)

Where,

  $n$ is number of half-lives.

Write the expression to the age.

  $t=n{T}_{\frac{1}{2}}$ .... (2)

Where,

  $T_{\frac{1}{2}}$ is the half-life of carbon-14.

Calculation:

Substitute the values in equation (1).

  $\begin{array}{c}0.060={\left(\frac{1}{2}\right)}^n\\ \log \left(0.060\right)=n\log \left(\frac{1}{2}\right)\\ n\approx 4.06\end{array}$

Substitute the values in equation (2).

  $\begin{array}{c}t=4.06\times 5730\\ =23263.8\text{years}\end{array}$

Conclusion:

Therefore, the age of the tool is $23263.8\text{years}$ .