Chapter 4.2, Problem 34PS

Solution

a.

To find: The percent of carbon-14 remains after 2500 years, 5000 years, and 10000 years.

73.9% of the original amount of carbon-14 remains in the sample after 2500 years.

54.6% of the original amount of carbon-14 remains in the sample after 5000 years.

29.8% of the original amount of carbon-14 remains in the sample after 10000 years.

Given:

The equation $P=100{\left(\frac{1}{2}\right)}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$5730$}\right.}$ gives the percent $P$ of the original amount of carbon-14 remains in the sample after $t$ years.

Calculation:

When $t=2500$ ,

  $\begin{array}{c}P=100{\left(\frac{1}{2}\right)}^{\raisebox{1ex}{$2500$}\!\left/ \!\raisebox{-1ex}{$5730$}\right.}\\ \approx 73.9\%\end{array}$

Hence, 73.9% of the original amount of carbon-14 remains in the sample after 2500 years.

When $t=5000$ ,

  $\begin{array}{c}P=100{\left(\frac{1}{2}\right)}^{\raisebox{1ex}{$5000$}\!\left/ \!\raisebox{-1ex}{$5730$}\right.}\\ \approx 54.6\%\end{array}$

Hence, 54.6% of the original amount of carbon-14 remains in the sample after 5000 years.

When $t=10000$ ,

  $\begin{array}{c}P=100{\left(\frac{1}{2}\right)}^{\raisebox{1ex}{$10000$}\!\left/ \!\raisebox{-1ex}{$5730$}\right.}\\ \approx 29.8\%\end{array}$

Hence, 29.8% of the original amount of carbon-14 remains in the sample after 10000 years.

b.

To graph: The given model.

  

Given:

The equation $P=100{\left(\frac{1}{2}\right)}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$5730$}\right.}$ gives the percent $P$ of the original amount of carbon-14 remains in the sample after $t$ years.

Calculation:

The given equation passes through the points $\left(0,100\right),\left(5730,50\right)$ .

Now the graph is,

  

c.

To find: The age of the bone when it was found.

8129 years.

Given:

The equation $P=100{\left(\frac{1}{2}\right)}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$5730$}\right.}$ gives the percent $P$ of the original amount of carbon-14 remains in the sample after $t$ years.

Calculation:

From the graph in subpart (b), it can be seen that 37% of the carbon-14 present in the sample after 8219 years.

Hence, the age of the bone is 8129 years when it was found.